Polynomial Functions

Chapter Overview and Pacing

PACING (days)
Regular Block
LESSON OBJECTIVES Basic/ Basic/
Average Advanced Average Advanced
Polynomial Functions (pp. 346–352) 1 1 0.5 0.5
• Evaluate polynomial functions.
• Identify general shapes of graphs of polynomial functions.
Graphing Polynomial Functions (pp. 353–359) 1 2 0.5 1.5
• Graph polynomial functions and locate their real zeros. (with 7-2 (with 7-2
• Find the maxima and minima of polynomial functions. Follow-Up) Follow-Up)
Follow-Up: Modeling Real-World Data
Solving Equations Using Quadratic Techniques (pp. 360–364) 2 2 1 1
• Write expressions in quadratic form.
• Use quadratic techniques to solve equations.
The Remainder and Factor Theorems (pp. 365–370) 2 2 1 1
• Evaluate functions using synthetic substitution.
• Determine whether a binomial is a factor of a polynomial by using synthetic substitution.
Roots and Zeros (pp. 371–377) 1 1 0.5 0.5
• Determine the number and type of roots for a polynomial equation.
• Find the zeros of a polynomial function.
Rational Zero Theorem (pp. 378–382) 2 2 1 1
• Identify the possible rational zeros of a polynomial function.
• Find all the rational zeros of a polynomial function.
Operations on Functions (pp. 383–389) 1 1 0.5 0.5
• Find the sum, difference, product, and quotient of functions.
• Find the composition of functions.
Inverse Functions and Relations (pp. 390–394) 1 1 0.5 0.5
• Find the inverse of a function or relation.
• Determine whether two functions or relations are inverses.
Square Root Functions and Inequalities (pp. 395–399) 1 1 0.5 0.5
• Graph and analyze square root functions.
• Graph square root inequalities.
Study Guide and Practice Test (pp. 400–405) 1 1 0.5 0.5
Standardized Test Practice (pp. 406–407)
Chapter Assessment 1 1 0.5 0.5
TOTAL 14 15 7 8

Pacing suggestions for the entire year can be found on pages T20–T21.

344A Chapter 7 Polynomial Functions

 Timesaving Tools
™

All-In-One Planner
and Resource Center
Chapter Resource Manager See pages T12–T13.

Chapter 7 RESOURCE MASTERS

ons al
ma Learn

ns*

ess utori
are heck
e)
ion
e

es
rag

lkb ve

)
nt
tics

atio
nt
Int Guid

d
and tice

nci
ent

T
C

Cha racti
oar
me
Ave

me
Ma ng to

Plu SS:
Tra inute
plic
erv

ess
ich
and tudy

(Sk Pra

A
the

s (l
nsp
Ap

Int

e2P
di

Enr

Ass

5-M
Rea
S

ills

Alg
Materials
375–376 377–378 379 380 7-1 7-1 graphing calculator, grid paper, string

381–382 383–384 385 386 SM 71–76 7-2 7-2 graphing calculator

(Follow-Up: graphing calculator)

387–388 389–390 391 392 443 7-3 7-3 13 colored pencils

393–394 395–396 397 398 7-4 7-4

399–400 401–402 403 404 443, 445 7-5 7-5 14 slips of paper

405–406 407–408 409 410 GCS 39 7-6 7-6

411–412 413–414 415 416 444 GCS 40, 7-7 7-7

SC 13

417–418 419–420 421 422 SC 14 7-8 7-8 grid paper, string, spaghetti

423–424 425–426 427 428 444 7-9 7-9 graphing calculator

429–442,
446–448

*Key to Abbreviations: GCS
Graphing Calculator and Speadsheet Masters,

SC
School-to-Career Masters,
SM
Science and Mathematics Lab Manual

Chapter 7 Polynomial Functions 344B

 Mathematical Connections
and Background
Continuity of Instruction Polynomial Functions
While this chapter uses some graphs to illustrate
ideas, the primary focus of the chapter is on algebraic
relationships between algebraic entities. In this lesson,
the algebraic entities are the terms of a polynomial func-
Prior Knowledge tion. The main algebraic relationship in the lesson is how
Students have evaluated polynomials and the degree and coefficient of one of those entities, the
terms of polynomials, graphed quadratic leading term of a polynomial function, determines the
end behavior of the function.
functions, and solved quadratic equations
The lesson begins by evaluating polynomial
and inequalities. They explored synthetic functions for various values of the variable, and uses
division and the equivalence between zeros the resulting sets of ordered pairs to illustrate the func-
of functions and roots of equations. Also, tions. Students use those illustrations to discuss when a
they worked with function notation and graph may or must cross the x-axis. The illustrations
manipulated and evaluated functions. also lead to the lesson’s primarily idea of end behavior:
for even-degree polynomial functions the left and right
extremes of the graph are both positive or both negative,
while for odd-degree polynomial functions one extreme
is positive and the other extreme is negative.

Graphing Polynomial Functions

This Chapter In this lesson, the primary algebraic entities are
Students look at algebraic relationships sets of x values, each set representing an interval on the
among algebraic entities. They use the lead- x-axis. These entities are used to explore two types of
ing term of a polynomial function to describe changes in the values of a polynomial function. One
type of change, a change between positive and negative
and identify the end behavior of the func- function values, indicates that the graph of the function
tion. They see how synthetic division of poly- is crossing the x-axis, which is where the function has a
nomials helps them explore the number of zero. The other type of change, a change between
roots of a polynomial function and then find increasing values and decreasing values for that x-axis
those roots. They perform operations on interval, is called a turning point of the function. A turn-
functions, and examine restrictions on ing point is a relative maximum or a relative minimum
of the function depending on whether the switch is from
domains and ranges necessary for increasing values to decreasing values, or vice versa.
combining functions. The lesson also uses the end-behavior analysis of the
previous lesson and the two types of changes to help
sketch graphs of polynomial functions.

Solving Equations Using

Future Connections Quadratic Techniques
Students will return to polynomials in later In this lesson, the algebraic entities are polyno-
mathematics and other courses. They will mial expressions that can be rewritten as quadratic trino-
mials or binomials; for example, a4  3a2  2 can be
continue their study of complex numbers, 2
rewritten as (a2)  3(a2)  2. Students rewrite polyno-
and extend it to study the complex plane. mial expressions whose exponents are integers or
They will frequently use functions, combina- rational numbers, and then factor or use the Quadratic
tions of functions, and inverses of functions Formula to solve related quadratic equations.
in future mathematics topics.

344C Chapter 7 Polynomial Functions

 The Remainder and then the fraction’s numerator and denominator must be
factors of a0 and an , respectively. Students use this prop-
Factor Theorems erty in a two-step process for finding all the rational
In this lesson, the algebraic entity is a statement zeros of a polynomial function. First, they find all possi-
f(x) k ble factors p of a0 , all possible factors q of an , and list
of polynomial division, 
q(x)   , where p
xa xa (and reduce) all possible fractions q. Second, they test
the right side of the equation is a polynomial plus a
fraction whose numerator is a constant and whose whether particular values are zeros. Once they find
denominator is the divisor. The lesson observes two zeros, they can also use the Remainder Theorem to
properties of polynomials. One property focuses on find the depressed polynomial for that zero.
the remainder. Multiplying both sides by the divisor
(x  a) and then finding the value of the polynomial at Operations on Functions
a gives the result f(a)
q(a)(a  a)  k. This simplifies In this lesson (and the next), the algebraic enti-
to f(a)
k. In words, if a polynomial is divided by ties are functions themselves. The algebraic relationship
(x  a), then the remainder k is equal to the value of is to look at the result of combining functions. Four
the polynomial at a, or f(a)
k. This property is called combinations interpret the arithmetic operations addi-
the Remainder Theorem. The other property focuses tion, subtraction, multiplication, and division for func-
on (x  a), a binomial that divides a polynomial f(x) tions. Another type of combination of functions is the
evenly. In such a case, (x  a) is a factor of f(x) and composition f
g, where [ f
g](x) = f [g(x)]. Students
the remainder must be zero; since f(a) is the value of also identify relationships between and restrictions
the remainder, f(a)
0. This property is called the on ranges and domains for combining functions.
Factor Theorem. Students use these properties, along
with the result that the degree of q(x) is one less than
the degree of f(x), to find all the factors of third-degree Inverse Functions
polynomials. and Relations
In this lesson, the algebraic entities are functions
Roots and Zeros and the relationship explored is that of inverse func-
This lesson focuses on the equivalent algebraic tions. Given a function f(x), students find another
entities of roots of a polynomial equation, zeros of func- function g(x) by switching the variables x and y and
tions, factors of polynomials, and x-intercepts of graphs. solving the resulting equation for y. Then they test
The lesson presents two properties. One property, called whether f(x) and g(x) are inverses by checking that each
the Fundamental Theorem of Algebra, states the exis- of the two compositions [ f
g](x) and [g
f ](x) has the
tence of at least one complex root for a polynomial value x. The lesson uses the notation f 1(x) for the
equation. Based on this property, students learn that inverse function of f(x) and the notation I(x) for the
the degree of a polynomial equation is the same as identify function, and introduces the term one-to-one
the number of its complex roots. The other property, for a function that passes the horizontal line test.
called Descartes’ Rule of Signs, lets students calculate
the number of positive and negative zeros of a poly- Square Root Functions
nomial equation. Then, given a polynomial equation, and Inequalities
students use the two properties to find out how many
roots they are looking for, how many will be real, and In this lesson, the algebraic entity is a square
how many (as complex conjugates) will be complex. root function, so called because the function contains a
With that information, they use methods from the variable inside a square root symbol. The lesson
previous lessons to find the roots of the polynomial explores two main ideas. One idea is that while the
equation. inverse of a quadratic function is not a function, you can
restrict the domain of the inverse so that the result is a
function. The other idea is that to graph a square root
Rational Zero Theorem inequality, first you graph the related square root equa-
In this lesson, the algebraic entities are the lead- tion, forming two regions. Then you check points to
ing term a0 xn and constant term an of a polynomial see which region is represented by the inequality, and
function with integer coefficients. The algebraic prop- you use the inequality symbol to decide whether or
erty is that if a reduced fraction is a zero of a function, not the boundary line is part of the solution region.

Chapter 7 Polynomial Functions 344D

 and Assessment

Type Student Edition Teacher Resources Technology/Internet

Ongoing Prerequisite Skills, pp. 345, 352, 5-Minute Check Transparencies Alge2PASS: Tutorial Plus
INTERVENTION

358, 364, 370, 377, 382, 389, Quizzes, CRM pp. 443–444 www.algebra2.com/self_check_quiz
394 Mid-Chapter Test, CRM p. 445 www.algebra2.com/extra_examples
Practice Quiz 1, p. 364 Study Guide and Intervention, CRM pp. 375–376,
Practice Quiz 2, p. 382 381–382, 387–388, 393–394, 399–400, 405–406,
411–412, 417–418, 423–424
Mixed pp. 352, 358, 364, 370, 377, Cumulative Review, CRM p. 446
Review 382, 389, 394, 399
Error Find the Error, pp. 380, 386 Find the Error, TWE pp. 380, 386
Analysis Unlocking Misconceptions, TWE pp. 354, 361, 375
Tips for New Teachers, TWE p. 384
Standardized pp. 352, 358, 364, 370, 374, TWE p. 374 Standardized Test Practice
Test Practice 375, 377, 382, 389, 394, 399, Standardized Test Practice, CRM pp. 447–448 CD-ROM
405, 406–407 www.algebra2.com/
standardized_test
Open-Ended Writing in Math, pp. 352, 357, Modeling: TWE pp. 352, 389, 394
Assessment 364, 370, 377, 382, 389, 394, Speaking: TWE pp. 364, 370, 382
399 Writing: TWE pp. 358, 377, 399
Open Ended, pp. 350, 356, 362, Open-Ended Assessment, CRM p. 441
368, 375, 380, 382, 386, 393,
ASSESSMENT

397
Chapter Study Guide, pp. 400–404 Multiple-Choice Tests (Forms 1, 2A, 2B), TestCheck and Worksheet Builder
Assessment Practice Test, p. 405 CRM pp. 429–434 (see below)
Free-Response Tests (Forms 2C, 2D, 3), MindJogger Videoquizzes
CRM pp. 435–440 www.algebra2.com/
Vocabulary Test/Review, CRM p. 442 vocabulary_review
www.algebra2.com/chapter_test

Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters

Additional Intervention Resources TestCheck and Worksheet Builder

The Princeton Review’s Cracking the SAT & PSAT This networkable software has three modules for intervention
The Princeton Review’s Cracking the ACT and assessment flexibility:
• Worksheet Builder to make worksheet and tests
ALEKS • Student Module to take tests on screen (optional)
• Management System to keep student records (optional)
Special banks are included for SAT, ACT, TIMSS, NAEP, and
End-of-Course tests.

344E Chapter 7 Polynomial Functions

 Reading and Writing
in Mathematics
Intervention Technology Glencoe Algebra 2 provides numerous opportunities to
Alge2PASS: Tutorial Plus CD-ROM offers a incorporate reading and writing into the mathematics
complete, self-paced algebra curriculum. classroom.

Algebra 2 Alge2PASS Lesson Student Edition

Lesson
• Foldables Study Organizer, p. 345
7-3 13 Graphing Polynomial Functions • Concept Check questions require students to verbalize
7-5 14 Finding Roots and Zeros and write about what they have learned in the lesson.
(pp. 350, 356, 362, 368, 375, 380, 386, 393, 397, 400)
ALEKS is an online mathematics learning system that • Writing in Math questions in every lesson, pp. 352, 357,
adapts assessment and tutoring to the student’s needs. 364, 370, 377, 382, 389, 394, 399
Subscribe at www.k12aleks.com. • Reading Study Tip, pp. 354, 372, 384, 391
• WebQuest, p. 399

Intervention at Home Teacher Wraparound Edition

• Foldables Study Organizer, pp. 345, 400
Log on for student study help. • Study Notebook suggestions, pp. 350, 356, 362, 368,
• For each lesson in the Student Edition, there are Extra 375, 380, 386, 392, 397
Examples and Self-Check Quizzes. • Modeling activities, pp. 352, 389, 394
www.algebra2.com/extra_examples • Speaking activities, pp. 364, 370, 382
www.algebra2.com/self_check_quiz • Writing activities, pp. 358, 377, 399
• For chapter review, there is vocabulary review, test • Differentiated Instruction, (Verbal/Linguistic), p. 356
practice, and standardized test practice. • ELL Resources, pp. 344, 351, 356, 357, 363, 369,
www.algebra2.com/vocabulary_review 376, 381, 388, 393, 398, 400
www.algebra2.com/chapter_test
www.algebra2.com/standardized_test Additional Resources
• Vocabulary Builder worksheets require students to
define and give examples for key vocabulary terms as
For more information on Intervention and they progress through the chapter. (Chapter 7 Resource
Assessment, see pp. T8–T11. Masters, pp. vii-viii)
• Reading to Learn Mathematics master for each lesson
(Chapter 7 Resource Masters, pp. 379, 385, 391, 397,
403, 409, 415, 421, 427)
• Vocabulary PuzzleMaker software creates crossword,
jumble, and word search puzzles using vocabulary lists
that you can customize.
• Teaching Mathematics with Foldables provides
suggestions for promoting cognition and language.
• Reading and Writing in the Mathematics Classroom
• WebQuest and Project Resources

For more information on Reading and Writing in

Mathematics, see pp. T6–T7.
Chapter 7 Polynomial Functions 344F
 Notes Polynomial
Functions
Have students read over the list
of objectives and make a list of
any words with which they are
not familiar. • Lessons 7-1 and 7-3 Evaluate polynomial
Key Vocabulary
functions and solve polynomial equations. • polynomial function (p. 347)
• Lessons 7-2 and 7-9 Graph polynomial and • synthetic substitution (p. 365)
square root functions. • Fundamental Theorem
• Lessons 7-4, 7-5, and 7-6 Find factors and zeros of Algebra (p. 371)
of polynomial functions. • composition of
Point out to students that this is • Lesson 7-7 Find the composition of functions. functions (p. 384)
only one of many reasons why • Lesson 7-8 Determine the inverses of functions • inverse function
each objective is important. or relations. (p. 391)
Others are provided in the
introduction to each lesson.

According to the Fundamental Theorem of Algebra, every polynomial

equation has at least one root. Sometimes the roots have real-world
meaning. Many real-world situations that cannot be modeled using a
linear function can be approximated using a polynomial function.
You will learn how the power generated by a windmill can be modeled by a
polynomial function in Lesson 7-1.

NCTM Local
Lesson Standards Objectives
7-1 1, 2, 6, 7, 8, 9,
10
7-2 1, 2, 6, 8, 9, 10
7-2 2, 5, 6, 9, 10
Follow-Up
7-3 1, 2, 3, 4, 6, 8,
9, 10
7-4 1, 2, 3, 6, 8, 9,
10
7-5 1, 2, 3, 4, 6, 7,
8, 9, 10
7-6 1, 2, 3, 4, 6, 7,
8, 9
7-7 1, 2, 6, 7, 8, 9,
10
344 Chapter 7 Polynomial Functions
7-8 1, 2, 3, 4, 6, 7,
8, 9, 10
7-9 1, 2, 6, 7, 8, 9,
10
Vocabulary Builder ELL

The Key Vocabulary list introduces students to some of the main vocabulary terms
Key to NCTM Standards: included in this chapter. For a more thorough vocabulary list with pronunciations of
1=Number & Operations, 2=Algebra, new words, give students the Vocabulary Builder worksheets found on pages vii and
3=Geometry, 4=Measurement, viii of the Chapter 7 Resource Masters. Encourage them to complete the definition
5=Data Analysis & Probability, 6=Problem of each term as they progress through the chapter. You may suggest that they add
Solving, 7=Reasoning & Proof, these sheets to their study notebooks for future reference when studying for the
8=Communication, 9=Connections, Chapter 7 test.
10=Representation
344 Chapter 7 Polynomial Functions
Prerequisite Skills To be successful in this chapter, you’ll need to master This section provides a review of
these skills and be able to apply them in problem-solving situations. Review the basic concepts needed before
these skills before beginning Chapter 7.
beginning Chapter 7. Page
For Lesson 7-2 3. between –5 and –4, between 0 and 1 Solve Equations by Graphing
references are included for
additional student help.
Use the related graph of each equation to determine its roots. If exact roots cannot be
found, state the consecutive integers between which the roots are located.
(For review, see Lesson 6-2.) 1. between 0 and 1, between 4 and 5 2. between
2 and
1, 1 Prerequisite Skills in the Getting
1. x2  5x  2 = 0 2. 3x2 x40
2
3.

x2  3x  1  0 Ready for the Next Lesson section
3
at the end of each exercise set
f (x ) f (x ) f (x )
review a skill needed in the next
lesson.
O x O x O x

For Prerequisite
f (x )  3x 2  x
4
Lesson Skill
f (x )  x 2
5 x  2 f (x ) 
2 2
x  3x
1 7-2 Graphing Quadratic Functions
3
(p. 352)
For Lesson 7-3 Quadratic Formula 7-3 Factoring Polynomials (p. 358)
Solve each equation. (For review, see Lesson 6-5.) 7-4 Dividing Polynomials (p. 364)

5
17
4. x2  17x  60  0 5, 12 5. 14x2  23x  3  0 3 16. 2x2  5x  1  0

7-5 Quadratic Formula (p. 370)



,


4
2 7
For Lessons 7-4 through 7-6 Synthetic Division
7-7 Operations with Polynomials
(p. 382)
Simplify each expression using synthetic division. (For review, see Lesson 5-3.)
33 7-8 Solving Equations for a
7. (3x2  14x  24)  (x  6) 3x  4 8. (a2  2a  30)  (a  7) a
9 


a7 Variable (p. 389)
For Lessons 7-1 and 7-7 Evaluating Functions 7-9 Solving Radical Equations
Find each value if f(x)  4x  7 and g(x)  2x2  3x  1. (For review, see Lesson 2-1.) (p. 394)
9. f(3)
19 10. g(2a) 8a2
6a  1 11. f(4b2)  g(b) 18b2
3b
6

Make this Foldable to help you organize information about

1
polynomial functions. Begin with five sheets of plain 8

" by
2
11" paper.
Stack and Fold Staple and Label

Stack sheets Staple along

of paper with fold. Label the s
Polynomial
edges 34 -inch apart. tabs with lesson
7-1
Fold up bottom numbers. 7-2
edges to create 7-3
7-4
equal tabs. 7-5
7-6
7-7
7-8
7-9

Reading and Writing As you read and study the chapter, use each page to write
notes and examples.

Chapter 7 Polynomial Functions 345

TM

Making Generalizations while Previewing and Reviewing Data

Have students label each tab of their Foldable to correspond to a
lesson in Chapter 7. Before reading each lesson, have students pre-
For more information view it and write generalizations about what they think they will learn.
about Foldables, see As they read and study each lesson, students use their Foldable to
Teaching Mathematics take notes, define terms, record concepts, and write examples.
with Foldables. After each lesson, ask them to compare what they thought they
would learn with what they did learn as they review their notes.

Chapter 7 Polynomial Functions 345

 Lesson Polynomial Functions
Notes

• Evaluate polynomial functions.

1 Focus • Identify general shapes of graphs of polynomial functions.

Vocabulary Where are polynomial functions found in nature?

5-Minute Check • polynomial in one If you look at a cross section of a honeycomb, you see a
Transparency 7-1 Use as a variable pattern of hexagons. This pattern has one hexagon
quiz or review of Chapter 6. • degree of a polynomial surrounded by six more hexagons. Surrounding these is
• leading coefficients a third ring of 12 hexagons, and so on. The total number
Mathematical Background notes • polynomial function of hexagons in a honeycomb can be modeled by the
are available for this lesson on • end behavior function f(r)  3r2  3r  1, where r is the number of
rings and f(r) is the number of hexagons.
p. 344C.

Where are polynomial func-

tions found in nature?
Study Tip
POLYNOMIAL FUNCTIONS Recall that a polynomial is a monomial or a sum
Look Back
Ask students: To review polynomials, of monomials. The expression 3r2  3r  1 is a polynomial in one variable since it
see Lesson 5-2. only contains one variable, r.
• How could you confirm that
this model works for r  1 and
r  2? Substitute 1 and 2 for r in the
Polynomial in One Variable
formula and compare the results • Words A polynomial of degree n in one variable x is an expression of the
form a0xn  a1xn  1  …  an  2x2  an  1x  an, where the
with the number of hexagons you coefficients a0, a1, a2, …, an, represent real numbers, a0 is not zero,
count in the photograph. and n represents a nonnegative integer.
• Using the formula, what is the • Examples 3x5  2x4  5x3  x2  1
value of f(8)? 169 n  5, a0  3, a1  2, a2  5, a3  1, a4  0, and a5  1

• Why might it be useful for a

beekeeper to know the approxi-
The degree of a polynomial in one variable is the greatest exponent of its variable.
mate number of hexagons in a The leading coefficient is the coefficient of the term with the highest degree.
honeycomb? Sample answer:
The amount of honey stored in a Polynomial Expression Degree Leading
Coefficient
honeycomb could be predicted by
Constant 9 0 9
the number of hexagons.
Linear x2 1 1
Quadratic 3x2  4x  5 2 3
Cubic 4x3  6 3 4
General a0xn  a1xn  1  …  an  2x 2  an  1x  an n a0

Example 1 Find Degree and Leading Coefficients

State the degree and leading coefficient of each polynomial in one variable. If it
is not a polynomial in one variable, explain why.
a. 7x4  5x2  x
9
This is a polynomial in one variable.
The degree is 4, and the leading coefficient is 7.
346 Chapter 7 Polynomial Functions

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 7 Resource Masters 5-Minute Check Transparency 7-1
• Study Guide and Intervention, pp. 375–376 Answer Key Transparencies
• Skills Practice, p. 377
• Practice, p. 378 Technology
• Reading to Learn Mathematics, p. 379 Interactive Chalkboard
• Enrichment, p. 380
 b. 8x2  3xy
2y2
This is not a polynomial in one variable. It contains two variables, x and y. 2 Teach
1
c. 7x6
4x3 


x
1
POLYNOMIAL FUNCTIONS
This is not a polynomial. The term

cannot be written in the form xn, where n
x
is a nonnegative integer. In-Class Examples Power
Point®
1
d.

x2  2x3
x5
2 Teaching Tip Stress that the
Rewrite the expression so the powers of x are in decreasing order. leading coefficient is not always
1 the coefficient of the first term
x5  2x3 

x2
Study Tip 2
of a polynomial.
This is a polynomial in one variable with degree of 5 and leading coefficient of 1.
Power Function
A common type of 1 State the degree and leading
function is a power
function, which has an
coefficient of each polynomial
A polynomial equation used to represent a function is called a polynomial
equation in the form function . For example, the equation f(x)  4x2  5x  2 is a quadratic polynomial in one variable. If it is not a
f(x)  axb, where a and b function, and the equation p(x)  2x3  4x2  5x  7 is a cubic polynomial function. polynomial in one variable,
are real numbers. When b
is a positive integer,
Other polynomial functions can be defined by the following general rule. explain why.
f(x)  axb is a polynomial a. 7z3  4z2  z degree 3,
function. Definition of a Polynomial Function leading coefficient 7
• Words A polynomial function of degree n can be described by an equation
of the form P(x)  a0xn  a1 xn  1  …  an  2x2  an  1x  an, b. 6a3  4a2  ab2 This is not a
where the coefficients a0, a1, a2, …, an, represent real numbers, polynomial in one variable. It
a0 is not zero, and n represents a nonnegative integer. contains two variables, a and b.
• Examples f(x)  4x2  3x  2 c. 3c2  4c  2c1 This is not a
n  2, a0  4, a1  3, a2  2
polynomial. The term
2c
1 is
not of the form anc n, where n is a
nonnegative integer.
If you know an element in the domain of any polynomial function, you can find
the corresponding value in the range. Recall that f(3) can be found by evaluating the d. 9y  3y2  y4 degree 4,
function for x  3. leading coefficient 1

Example 2 Evaluate a Polynomial Function 2 NATURE Refer to the appli-

NATURE Refer to the application at the beginning of the lesson. cation at the beginning of
a. Show that the polynomial function f(r)  3r2
3r  1 gives the total number
the lesson. A sketch of the
ring 3
of hexagons when r  1, 2, and 3. arrangement of hexagons
Find the values of f(1), f(2), and f(3).
shows a fourth ring of
ring 2
f(r)  3r2  3r  1 f(r)  3r2  3r  1 f(r)  3r2  3r  1
18 hexagons, a fifth ring of
ring 1
f(1)  3(1)  3(1)  1
2 f(2)  3(2)  3(2)  1
2 f(3)  3(3)2  3(3)  1
24 hexagons, and a sixth ring
 3  3  1 or 1  12  6  1 or 7  27  9  1 or 19
of 30 hexagons.
From the information given, you know the number of hexagons in the first a. Show that the polynomial
ring is 1, the number of hexagons in the second ring is 6, and the number of function f(r)  3r2  3r  1
hexagons in the third ring is 12. So, the total number of hexagons with one ring gives the total number of
is 1, two rings is 6  1 or 7, and three rings is 12  6  1 or 19. These match the
Rings of a Honeycomb
functional values for r  1, 2, and 3, respectively. hexagons when r  4, 5, and 6.
f(4)  48
12  1, or 37;
b. Find the total number of hexagons in a honeycomb with 12 rings. f(5)  75
15  1, or 61;
f(r)  3r2  3r  1 Original function f(6)  108
18  1, or 91; The
f(12)  3(12)2  3(12)  1 Replace r with 12. total number of hexagons for four
 432  36  1 or 397 Simplify. rings is 19  18 or 37, five rings
is 37  24 or 61, and six rings is
www.algebra2.com/extra_examples Lesson 7-1 Polynomial Functions 347 61  30 or 91. These match the
functional values for r  4, 5,
and 6, respectively.
b. Find the total number of
hexagons in a honeycomb
with 20 rings. 1141

Lesson 7-1 Polynomial Functions 347

In-Class Example Power
Point®
You can also evaluate functions for variables and algebraic expressions.

Example 3 Functional Values of Variables

3 a. Find p(y3) if
a. Find p(a2) if p(x)  x3  4x2
5x.
p(x)  2x4  x3  3x.
2y 12
y 9  3y 3 p(x)  x3  4x2  5x Original function
p(a2)  (a2)3  4(a2)2  5(a2) Replace x with a2.
b. Find b(2x  1)  3b(x) if
 a6  4a4  5a2 Property of powers
b(m)  2m2  m  1.
2x 2
9x  3 b. Find q(a  1)
2q(a) if q(x)  x2  3x  4.
To evaluate q(a  1), replace x in q(x) with a  1.
q(x)  x2  3x  4 Original function

GRAPHS OF POLYNOMIAL q(a  1)  (a  1)2  3(a  1)  4 Replace x with a + 1.

FUNCTIONS  a2  2a  1  3a  3  4 Evaluate (a + 1)2 and 3(a + 1).

 a2  5a  8 Simplify.
Teaching Tip Since the graphs on To evaluate 2q(a), replace x with a in q(x), then multiply the expression by 2.
this page show the maximum num- q(x)  x2  3x  4 Original function
ber of times each type of graph may 2q(a)  2(a2  3a  4) Replace x with a.
intersect the x-axis, some students  2a2  6a  8 Distributive Property
may ask about the minimum number Now evaluate q(a  1)  2q(a).
of times each graph type may inter- q(a  1)  2q(a)  a2  5a  8  (2a2  6a  8) Replace q(a + 1) and 2q(a)
sect the x-axis. Have students work in with evaluated expressions.
 a2  5a  8  2a2  6a  8
pairs using the given graphs to dis-
 a2  a Simplify.
cuss this issue. Lead students to see
that for functions of degree 1 the
minimum is 1 (the same as the maxi-
mum), for functions of degree 2 the GRAPHS OF POLYNOMIAL FUNCTIONS The general shapes of the
graphs of several polynomial functions are shown below. These graphs show the
minimum is 0, for functions of degree maximum number of times the graph of each type of polynomial may intersect the
3 the minimum is 1, for functions of x-axis. Recall that the x-coordinate of the point at which the graph intersects the
degree 4 the minimum is 0, and for x-axis is called a zero of a function. How does the degree compare to the
functions of degree 5 the minimum maximum number of real zeros?
is 1. Some students may notice the Constant function Linear function Quadratic function
pattern for functions with odd and Degree 0 Degree 1 Degree 2
even degrees. f (x ) f (x ) f (x )

O
O x O x x

Cubic function Quartic function Quintic function

Degree 3 Degree 4 Degree 5

f (x ) f (x ) f (x )

O x O x O x

Interactive
Chalkboard Notice the shapes of the graphs for even-degree polynomial functions and odd-
degree polynomial functions. The degree and leading coefficient of a polynomial
PowerPoint®
Presentations
function determine the graph’s end behavior.
348 Chapter 7 Polynomial Functions
This CD-ROM is a customizable
Microsoft® PowerPoint®
presentation that includes:
• Step-by-step, dynamic solutions of
each In-Class Example from the
Teacher Wraparound Edition
• Additional, Your Turn exercises for
each example
• The 5-Minute Check Transparencies
• Hot links to Glencoe Online
Study Tools

348 Chapter 7 Polynomial Functions

 The end behavior is the behavior of the graph as x approaches positive infinity
() or negative infinity (). This is represented as x →  and x → , In-Class Example Power
Point®
respectively. x →  is read x approaches positive infinity.
4 For each graph,
End Behavior of a Polynomial Function • describe the end behavior,
Degree: even Degree: odd Degree: even Degree: odd • determine whether it repre-
Leading Leading Leading Leading sents an odd-degree or an
Coefficient: positive Coefficient: positive Coefficient: negative Coefficient: negative
End Behavior: End Behavior: End Behavior: End Behavior: even-degree polynomial
f (x ) ∞ f (x ) ∞ f (x ) ∞ f (x ) ∞ function, and
as x
∞ as x ∞ as x ∞ as x
∞ • state the number of real
f (x ) f (x ) f (x )
f (x ) zeros.
f (x ) 
x 2
f (x ) 
x 3 a.
f (x )  x 3 f (x )
O
O x O x x O x
f (x )  x 2

f (x )
∞ f (x )
∞ f (x )
∞ f (x )
∞
O x
as x
∞ as x
∞ as x ∞ as x ∞

The graph of an even-degree function may or may not intersect the x-axis,
depending on its location in the coordinate plane. If it intersects the x-axis in two
Study Tip places, the function has two real zeros. If it does not intersect the x-axis, the roots of
Number of Zeros
the related equation are imaginary and cannot be determined from the graph. If the • f(x) →
 as x → .
graph is tangent to the x-axis, as shown above, there are two zeros that are the same f(x) →
 as x →
.
The number of zeros of an
odd-degree function may
number. The graph of an odd-degree function always crosses the x-axis at least once,
and thus the function always has at least one real zero. • It is an even-degree
be less than the maximum
by a multiple of 2. For polynomial function.
example, the graph of a Example 4 Graphs of Polynomial Functions • The graph does not intersect
quintic function may only
cross the x-axis 3 times.
For each graph, the x-axis, so the function has
• describe the end behavior, no real zeros.
f (x )
• determine whether it represents an odd-degree or an even-degree polynomial b. f (x )
function, and
• state the number of real zeros.
O x a. f (x ) b. f (x ) c. f (x )

O x
The same is true for an O x O x O x
even-degree function. One
exception is when the
graph of f(x) touches the
x-axis.
a. • f(x) →  as x → . f(x) →  as x → .
• f(x) →  as x → .
• It is an even-degree polynomial function. f(x) →
 as x →
.
• The graph intersects the x-axis at two points, so the function has two real • It is an odd-degree polynomial
zeros.
function.
b. • f(x) →  as x → . f(x) →  as x → .
• The graph intersects the x-axis
• It is an odd-degree polynomial function. at one point, so the function
• The graph has one real zero. has one real zero.
c. • f(x) →  as x → . f(x) →  as x → .
c. f (x )
• It is an even-degree polynomial function.
• This graph does not intersect the x-axis, so the function has no real zeros.

Lesson 7-1 Polynomial Functions 349

O x

Differentiated Instruction
Interpersonal Arrange students in groups of 3 or 4, providing each group
• f(x) →  as x → .
with a graphing calculator. Have each student write a polynomial function.
As a group, have students state whether each function is an odd-degree or f(x) →  as x →
.
an even-degree polynomial function before predicting the end behavior • It is an even-degree
and the number of zeros of the function. Then have students check their polynomial function.
predictions by graphing each function. Challenge students to find at least • The function has two real
one polynomial function that crosses the x-axis 3 or 4 times. zeros.

Lesson 7-1 Polynomial Functions 349

 3 Practice/Apply Concept Check 1. Explain why a constant polynomial such as f(x)  4 has degree 0 and a linear
4. Sometimes; a poly- polynomial such as f(x)  x  5 has degree 1. 4  4x 0; x  x 1
nomial function with 4 2. Describe the characteristics of the graphs of odd-degree and even-degree
real roots may be a polynomial functions whose leading coefficients are positive. See margin.
Study Notebook sixth-degree polynomial
3. OPEN ENDED Sketch the graph of an odd-degree polynomial function with a
function with 2 imaginary
Have students— roots. A polynomial negative leading coefficient and three real roots. See margin.
• add the definitions/examples of function that has 4 real 4. Tell whether the following statement is always, sometimes or never true. Explain.
roots is at least a fourth- A polynomial function that has four real roots is a fourth-degree polynomial.
the vocabulary terms to their degree polynomial.
Vocabulary Builder worksheets for Guided Practice State the degree and leading coefficient of each polynomial in one variable. If it is
Chapter 7. not a polynomial in one variable, explain why.
GUIDED PRACTICE KEY 5. 5x6  8x2 6; 5 6. 2b  4b3  3b5  7 5;
3
• copy the information in the Concept
Exercises Examples
Summary about end behavior Find p(3) and p(
1) for each function.
5, 6 1
shown on p. 349. 7. p(x)  x3  x2  x
21; 3 8. p(x)  x4  3x3  2x2  5x  1 4; 12
7, 8, 15 2
• include any other item(s) that they 9
11 3 If p(x)  2x3  6x
12 and q(x)  5x2  4, find each value. 11. 6a 3
5a 2  8a
45
12
14 4
find helpful in mastering the skills 9. p(a3) 2a 9  6a 3
12 10. 5[q(2a)] 100a 2  20 11. 3p(a)  q(a  1)
in this lesson.
12. a. f (x) →
∞ as For each graph,
x → ∞, f (x) → ∞ a. describe the end behavior,
as x →
∞; b. odd; b. determine whether it represents an odd-degree or an even-degree polynomial
c. 3 function, and
13. a. f (x) → ∞ as c. state the number of real zeros.
x → ∞, f (x) → ∞
12. 13. 14.
About the Exercises… as x →
∞; b. even; f (x ) f (x ) f (x )
c. 0
Organization by Objective
14. a. f (x) → ∞ as
• Polynomial Functions: 16–38 x → ∞, f (x) →
∞ O x O x O x
• Graphs of Polynomial as x →
∞; b. odd;
Functions: 39–44 c. 1
Odd/Even Assignments Application 15. BIOLOGY The intensity of light emitted by a firefly can be determined by
Exercises 16–44 are structured L(t)  10  0.3t  0.4t2  0.01t3, where t is temperature in degrees Celsius and
so that students practice the L(t) is light intensity in lumens. If the temperature is 30°C, find the light
same concepts whether they intensity. 109 lumens
are assigned odd or even ★ indicates increased difficulty
problems. Practice and Apply
Homework Help State the degree and leading coefficient of each polynomial in one variable. If it is
Assignment Guide For See not a polynomial in one variable, explain why.
Exercises Examples
Basic: 17–45 odd, 49–52, 56–70 16–21 1
16. 7  x 1;
1 17. (a  1)(a2  4) 3; 1
Average: 17–45 odd, 46–52, 22–29, 45 2 18. a2  2ab  b2 See margin. 19. 6x4  3x2  4x  8 4; 6
30–38 3
56–70 39–44, 4 20. 7  3x2  5x3  6x2  2x 3;
5
1
21. c2  c 

See margin.
46–48 c
Advanced: 16–44 even, 46–67 Find p(4) and p(
2) for each function.
(optional: 68–70) Extra Practice
See page 842. 22. p(x)  2  x
2; 4 23. p(x)  x2  3x  8 12; 18
24. p(x)  2x3  x2  5x  7 125;
37 25. p(x)  x5  x2 1008;
36
26. p(x)  x4  7x3  8x  6
166; 50 27. p(x)  7x2  9x  10 86; 56
Answers 1
28. p(x) 

x4  2x2  4 100; 4 1 1 1
29. p(x) 

x3 

x2 

x  5 7; 4
2 8 4 2
2. Sample answer: Even-degree 350 Chapter 7 Polynomial Functions

polynomial functions with positive

leading coefficients have graphs
in which f(x) →  as x →  3. Sample answer: 18. No, the polynomial contains two variables, a and b.
1
and as x →
. Odd-degree f (x) 21. No, this is not a polynomial because the term  cannot be
c
polynomial functions with positive written in the form x n, where n is a nonnegative integer.
leading coefficients have graphs 39a. f(x) →  as x → , f(x) →
 as x →
; b. odd; c. 3
in which f(x) →  as x →  40a. f(x) →  as x → , f(x) →  as x →
; b. even; c. 4
and f(x) →
 as x →
. O x
41a. f(x) →
 as x → , f(x) →
 as x →
; b. even; c. 0
42a. f(x) →  as x → , f(x) →
 as x →
; b. odd; c. 5
43a. f(x) →  as x → , f(x) →
 as x →
; b. odd; c. 1
44a. f(x) →
 as x → , f(x) →
 as x →
; b. even; c. 2
350 Chapter 7 Polynomial Functions
32. 3a4
2a2  5 If p(x)  3x2
2x  5 and r(x)  x3  x  1, find each value. Study
NAME ______________________________________________ DATE

Guide andIntervention
Intervention,
____________ PERIOD _____

7-1 Study Guide and

34. x3  3x2  4x  3 30. r(3a) 27a3  3a  1 31. 4p(a) 12a2
8a  20 32. p(a2) p. 375 (shown)
Polynomial Functions and p. 376

35. 3x4  16x2  26 33. p(2a3) 12a6
4a3  5 34. r(x  1) 35. p(x2  3) Polynomial Functions
A polynomial of degree n in one variable x is an expression of the form

36. 2[p(x  4)] 37. r(x  1)  r(x2) 38. 3[p(x2  1)]  4p(x)
Polynomial in a0x n  a1x n  1  …  an  2x 2  an  1x  an,
One Variable where the coefficients a0, a1, a2, …, an represent real numbers, a0 is not zero,
and n represents a nonnegative integer.

6x2  44x  90
x6  x3  2x2  4x  2 9x4
12x2
8x  50 The degree of a polynomial in one variable is the greatest exponent of its variable. The
leading coefficient is the coefficient of the term with the highest degree.

Lesson 7-1
For each graph, Polynomial
A polynomial function of degree n can be described by an equation of the form
P(x)  a0x n  a1x n  1  …  an  2x 2  an  1x  an,
Function where the coefficients a0, a1, a2, …, an represent real numbers, a0 is not zero,

a. describe the end behavior, and n represents a nonnegative integer.

Example 1
b. determine whether it represents an odd-degree or an even-degree polynomial What are the degree and leading coefficient of 3x2
2x4
7  x3 ?
Rewrite the expression so the powers of x are in decreasing order.
function, and 2x4  x3  3x2  7
This is a polynomial in one variable. The degree is 4, and the leading coefficient is 2.

c. state the number of real zeros. 39–44. See margin. Example 2 Find f(
5) if f(x)  x3  2x2
10x  20.
f(x)  x3  2x2  10x  20 Original function

39. f (x ) 40. f (x ) 41. f (x )

f(5)  (5)3  2(5)2  10(5)  20
 125  50  50  20
Replace x with 5.
Evaluate.
 5 Simplify.

Example 3 Find g(a2
1) if g(x)  x2  3x
4.

g(x)  x2  3x  4 Original function
g(a2  1)  (a2  1)2  3(a2  1)  4 Replace x with a 2  1.
 a4  2a2  1  3a2  3  4 Evaluate.
O x O x O x  a4  a2  6 Simplify.

Exercises
State the degree and leading coefficient of each polynomial in one variable. If it is
not a polynomial in one variable, explain why.
8; 8
1. 3x4  6x3  x2  12 4; 3 2. 100  5x3  10x7 7; 10 3. 4x6  6x4  8x8  10x2  20
x2 x6 x3 1
4. 4x2  3xy  16y2 5. 8x3  9x5  4x2  36 6.




18 25 36 72
42. f (x ) 43. f (x ) 44. f (x ) not a polynomial in
one variable; contains
5;
9
6;

1
25
two variables
Find f(2) and f(
5) for each function.

7. f(x)  x2  9 8. f(x)  4x3  3x2  2x  1 9. f(x)  9x3  4x2  5x  7

5; 16 23;
586 73;
1243
O x O x O x Gl NAME
/M G ______________________________________________
Hill 375 DATE ____________
GlPERIOD
Al _____
b 2

Skills
7-1 Practice,
Practice (Average)
p. 377 and
Practice,
Polynomialp. 378 (shown)
Functions
State the degree and leading coefficient of each polynomial in one variable. If it is
not a polynomial in one variable, explain why.
1 3 4 1
1. (3x2  1)(2x2  9) 4; 6 2.

a3 

a2 

a 3; 
5 5 5 5
Theater 2
3.
2  3m  12 Not a polynomial;
m
4. 27  3xy3  12x2y2  10y
45. ENERGY The power generated by a windmill is a function of the speed of the 2
 cannot be written in the form No, this polynomial contains two
In 1997, Cats surpassed A s3
m2
mn for a nonnegative integer n. variables, x and y.

Chorus Line as the longest- wind. The approximate power is given by the function P(s) 

, where s
1000 Find p(
2) and p(3) for each function.
running Broadway show. represents the speed of the wind in kilometers per hour. Find the units of power 5. p(x)  x3  x5 6. p(x)  7x2  5x  9 7. p(x)  x5  4x3
24;
216
29;
39 0;
135
Source: www.newsherald.com P(s) generated by a windmill when the wind speed is 18 kilometers per hour. 1 1 1 2
8. p(x)  3x3  x2  2x  5 9. p(x)  x4 

x3 

x 10. p(x) 

x3 

x2  3x
2 2 3 3

5.832 units
37; 73 13; 93
6; 24

If p(x)  3x2
4 and r(x)  2x2
5x  1, find each value.

THEATER For Exercises 46–48, use the 11. p(8a)

192a2
4
12. r(a2)
2a4
5a2  1
13. 5r(2a)

40a2  50a
5
Broadway Plays
graph that models the attendance to 14. r(x  2) 15. p(x2  1) 16. 5[p(x  2)]
2x2  3x
1 3x4
6x2
1 15x2  60x  40
Broadway plays (in millions) from 12
For each graph,
1970
2000.
Attendance (millions)

11 a. describe the end behavior,

b. determine whether it represents an odd-degree or an even-degree polynomial

46. Is the graph an odd-degree or 10 function, and

c. state the number of real zeroes.
9
even-degree function? even 17. f (x ) 18. f (x ) 19. f (x )

8
47. f (x) →
∞ as 47. Discuss the end behavior of the 7 O x O x O x

x → ∞; f (x) →
∞ graph. 6

as x →
∞ 48. Do you think attendance at 5 f(x) →  as x → , f(x) →  as x → , f(x) →  as x → ,
48. Sample answer: Broadway plays will increase or f(x) →  as x →
;
even; 2
f(x) →  as x →
;
even; 1
f(x) →
 as x →
;
odd; 5
Decrease; the graph 0 3 6 9 12 15 18 21 24 27 30
decrease after 2000? Explain your 20. WIND CHILL The function C(s)  0.013s2  s  7 estimates the wind chill temperature

appears to be turning reasoning. Years Since 1970 C(s) at 0F for wind speeds s from 5 to 30 miles per hour. Estimate the wind chill
temperature at 0F if the wind speed is 20 miles per hour. about
22F

at x  30 indicating a Gl NAME
/M G ______________________________________________
Hill 378 DATE ____________
Gl PERIOD
Al _____
b 2

relative maximum Reading

7-1 Readingto
to Learn
Learn Mathematics
at that point. So CRITICAL THINKING For Exercises 49–52, use the following information. Mathematics, p. 379 ELL
Polynomial Functions
attendance will The graph of the polynomial function f(x)  ax(x  4)(x  1) goes through the point Pre-Activity Where are polynomial functions found in nature?
Read the introduction to Lesson 7-1 at the top of page 346 in your textbook.
decrease after 2000. at (5, 15). • In the honeycomb cross section shown in your textbook, there is 1 hexagon
1 in the center, 6 hexagons in the second ring, and 12 hexagons in the third
49. Find the value of a.

ring. How many hexagons will there be in the fourth, fifth, and sixth rings?
2 18; 24; 30

50. For what value(s) of x will f(x)  0?
1, 0, 4 • There is 1 hexagon in a honeycomb with 1 ring. There are 7 hexagons in
a honeycomb with 2 rings. How many hexagons are there in honeycombs

Lesson 7-1
1 3 with 3 rings, 4 rings, 5 rings, and 6 rings?

51. Rewrite the function as a cubic function. f (x) 

x3


x2
2x 19; 37; 61; 91
2 2
52. Sketch the graph of the function. See margin. Reading the Lesson
1. Give the degree and leading coefficient of each polynomial in one variable.
www.algebra2.com/self_check_quiz Lesson 7-1 Polynomial Functions 351 degree leading coefficient
a. 10x3  3x2  x  7 3 10
b. 7y2  2y5  y  4y3 5
2
c. 100 0 100

NAME ______________________________________________ DATE ____________ PERIOD _____ 2. Match each description of a polynomial function from the list on the left with the
52. f (x )
corresponding end behavior from the list on the right.

8 Enrichment,
7-1 Enrichment p. 380 a. even degree, negative leading coefficient iii i. f(x) →  as x → ;
f(x) →  as x → 

b. odd degree, positive leading coefficient iv ii. f(x) →  as x → ;

f (x)  12 x 3
3 2
4 2 x
2x
Approximation by Means of Polynomials
Many scientific experiments produce pairs of numbers [x, f (x)] that can
f(x) →  as x → 

c. odd degree, negative leading coefficient ii iii. f(x) →  as x → ;

be related by a formula. If the pairs form a function, you can fit a
polynomial to the pairs in exactly one way. Consider the pairs given by f(x) →  as x → 
the following table. d. even degree, positive leading coefficient i iv. f(x) →  as x → ;
f(x) →  as x → 

4
2 x x 1 2 4 7
O 2 f (x) 6 11 39 54

We will assume the polynomial is of degree three. Substitute the given Helping You Remember

4 values into this expression. 3. What is an easy way to remember the difference between the end behavior of the graphs
of even-degree and odd-degree polynomial functions?
f(x)  A  B(x  x0)  C(x  x0)(x  x1)  D(x  x0)(x  x1)(x  x2)
Sample answer: Both ends of the graph of an even-degree function
You will get the system of equations shown below. You can solve this system

8 and use the values for A, B, C, and D to find the desired polynomial.
eventually keep going in the same direction. For odd-degree functions,
the two ends eventually head in opposite directions, one upward, the
6A other downward.
11  A  B(2  1)  A  B
39  A  B(4  1)  C(4  1)(4  2)  A  3B  6C
54  A  B(7  1)  C(7  1)(7  2)  D(7  1)(7  2)(7  4)  A  6B  30C  90D

Lesson 7-1 Polynomial Functions 351

 PATTERNS For Exercises 53–55, use the diagrams below that show the maximum

4 Assess number of regions formed by connecting points on a circle.

1 point, 1 region 2 points, 2 regions 3 points, 4 regions 4 points, 8 regions

Open-Ended Assessment
Modeling Provide students with
grid paper and a length of string.
Describe the end behavior and
number of real zeros of the graph 53. The maximum number of regions formed by connecting n points of a circle can
1
of a function and have students be described by the function f(n) 

(n4  6n3  23n2  18n  24). What is the
24
use their string to model a degree of this polynomial function? 4
possible graph that exhibits these ★ 54. Find the maximum number of regions formed by connecting 5 points of a circle.
characteristics. Draw a diagram to verify your solution. 16 regions; See margin for diagram.
★ 55. How many points would you have to connect to form 99 regions? 8 points

Getting Ready for 56. WRITING IN MATH Answer the question that was posed at the beginning of
the lesson. See margin.
Lesson 7-2 Where are polynomial functions found in nature?
PREREQUISITE SKILL In Lesson Include the following in your answer:
7-2, students will graph polyno- • an explanation of how you could use the equation to find the number of
mial functions by making a table hexagons in the tenth ring, and
of values. It is important that stu- • any other examples of patterns found in nature that might be modeled by a
dents know how to make tables polynomial equation.
of values and how to use them to Standardized 57. The figure at the right shows the graph of the f (x )
graph equations. Use Exercises Test Practice polynomial function f(x). Which of the following
68–70 to determine your students’ could be the degree of f(x)? C
familiarity with graphing A 2 B 3 C 4 D 5
quadratic functions by making a 1
O x

table of values. 58. If

x2  6x  2  0, then x could equal which of the

2
following? C
A 1.84 B 0.81 C 0.34 D 2.37
Answers
54. Maintain Your Skills
Mixed Review Solve each inequality algebraically. (Lesson 6-7)
59. {x2  x  6} 59. x2  8x  12  0 60. x2  2x  86 23
61. 15x2  3x  12
0
60. {xx
9 or
x
7}
Graph each function. (Lesson 6-6) 62–64. See margin.
x
1 x


4
5  
1 1 3
62. y  2(x  2)2  3 63. y 

(x  5)2  1 64. y 

x2  x 


3 2 2
56. Many relationships in nature can
Solve each equation by completing the square. (Lesson 6-4)
be modeled by polynomial
functions; for example, the 65. x2  8x  2  0 4  3
2
 1
3
35 7 5
66. x2 

x 

 0


,


36 6 6  
pattern in a honeycomb or the 67. 23,450(1  p); 67. BUSINESS Becca is writing a computer program to find the salaries of her
rings in a tree trunk. Answers 23,450(1  p)3 employees after their annual raise. The percent of increase is represented by p.
should include the following. Marty’s salary is $23,450 now. Write a polynomial to represent Marty’s salary
after one year and another to represent Marty’s salary after three years. Assume
• You can use the equation to find that the rate of increase will be the same for each of the three years. (Lesson 5-2)
the number of hexagons in a
honeycomb with 10 rings and Getting Ready for PREREQUISITE SKILL Graph each equation by making a table of values.
the number of hexagons in a the Next Lesson (To review graphing quadratic functions, see Lesson 6-1.)
1
68–70. See 68. y  x2  4 69. y  x2  6x  5 70. y 

x2  2x  6
honeycomb with 9 rings. The pp. 407A–407H. 2
difference is the number of 352 Chapter 7 Polynomial Functions

hexagons in the tenth ring.

• Other examples of patterns
62. y 63. y 64. y
found in nature include
y 
2(x
2)2  3
pinecones, pineapples, and
flower petals. 2

12
8 O x O x
O x

2 y  1 x2  x  3
2 2
y  1 (x  5)2
1
3
4

352 Chapter 7 Polynomial Functions

 Graphing Polynomial Lesson
Functions Notes

• Graph polynomial functions and locate their real zeros.

• Find the maxima and minima of polynomial functions.
1 Focus
Vocabulary can graphs of Foreign-Born Population
• Location Principle polynomial functions 5-Minute Check
P (t )
• relative maximum show trends in data? 18 Transparency 7-2 Use as a
• relative minimum 16
quiz or review of Lesson 7-1.

Percent of U.S.
14

Population
The percent of the United States
12
population that was foreign-born 10 Mathematical Background notes
since 1900 can be modeled by 8
P(t)  0.00006t3  0.007t2  0.05t  14, 6
are available for this lesson on
where t  0 in 1900. Notice that the 4 p. 344C.
2
graph is decreasing from t  5 to
t  75 and then it begins to increase. t
The points at t  5 and t  75 are
0 20 40 60 80 Building on Prior
Years Since 1900
turning points in the graph. Knowledge
In Chapter 6, students learned to
graph quadratic functions. Those
GRAPH POLYNOMIAL FUNCTIONS To graph a polynomial function, make
a table of values to find several points and then connect them to make a smooth same skills will be used in this
curve. Knowing the end behavior of the graph will assist you in completing the lesson to graph polynomial
sketch of the graph. functions.
can graphs of polyno-
Example 1 Graph a Polynomial Function
mial functions show
Study Tip Graph f(x)  x4  x3
4x2
4x by making a table of values. trends in data?
Graphing x f (x) x f (x) f (x ) Ask students:
Polynomial
2.5
8.4 0.0 0.0 • When the graph is sloping
Functions
To graph polynomial 2.0 0.0 0.5
2.8 downward to the right, what
O x
functions it will often 1.5
1.3 1.0 6.0
be necessary to include
does that tell you about the
x values that are not 1.0 0.0 1.5
6.6 population it represents? The
integers. 0.5
0.9 2.0 0.0 percent of the U.S. population that
f (x )  x 4  x 3
4 x 2
4 x is foreign-born is decreasing
during that span of time.
This is an even-degree polynomial with a positive leading coefficient, so f(x) → ∞
• If the United States government
as x → ∞, and f(x) → ∞ as x → ∞. Notice that the graph intersects the x-axis at banned any further immigra-
four points, indicating there are four real zeros of this function. tion, what would happen to
the graph? It would gradually
In Example 1, the zeros occur at integral values that can be seen in the table used
approach the horizontal axis.
to plot the function. Notice that the values of the function before and after each zero • Why would the graph not
are different in sign. In general, the graph of a polynomial function will cross the immediately reach the hori-
x-axis somewhere between pairs of x values at which the corresponding f(x) values
change signs. Since zeros of the function are located at x-intercepts, there is a zero zontal axis, where P(t)  0?
between each pair of these x values. This property for locating zeros is called the All of the current foreign-born
Location Principle . residents of the U.S. may still be
Lesson 7-2 Graphing Polynomial Functions 353 part of the population.

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 7 Resource Masters Science and Mathematics Lab Manual, 5-Minute Check Transparency 7-2
• Study Guide and Intervention, pp. 381–382 pp. 71–76 Answer Key Transparencies
• Skills Practice, p. 383
• Practice, p. 384 Technology
• Reading to Learn Mathematics, p. 385 Interactive Chalkboard
• Enrichment, p. 386

Lesson x-x Lesson Title 353

 Location Principle
2 Teach • Words Suppose y  f(x) represents a
polynomial function and a
• Model
f (b )
f (x )
(b , f (b ))

and b are two numbers such

GRAPH POLYNOMIAL that f(a)  0 and f(b)  0. O a
FUNCTIONS Then the function has at
f (a )
b x
least one real zero between (a , f (a ))
a and b.
In-Class Examples Power
Point®

1 Graph f(x)  x3  4x2  5

by making a table of values. Example 2 Locate Zeros of a Function
x f(x) Determine consecutive values of x between which each real zero of the function

4 5 f(x)  x3
5x2  3x  2 is located. Then draw the graph.
Make a table of values. Since f(x) is a third-degree polynomial function, it will have

3
4 either 1 or 3 real zeros. Look at the values of f(x) to locate the zeros. Then use the

2
3 points to sketch a graph of the function.

1 2 x f (x)
f (x )

0 5 2 32
1 7
1 0 0 2 } change in signs O x

19
2 1
2
1
4 } change in signs
f (x ) 3 7 f (x )  x 3
5 x 2  3 x  2
2
4
5 17 } change in signs

The changes in sign indicate that there are zeros between x  1 and x  0,
between x  1 and x  2, and between x  4 and x  5.
O x

MAXIMUM AND MINIMUM POINTS The

graph at the right shows the shape of a general third- f (x ) A relative
degree polynomial function. maximum
f (x) 
x 3
4x2 5
Point A on the graph is a relative maximum of the
cubic function since no other nearby points have a greater
2 Determine consecutive y-coordinate. Likewise, point B is a relative minimum B
values of x between which since no other nearby points have a lesser y-coordinate. relative O x
These points are often referred to as turning points. The minimum
each real zero of the function graph of a polynomial function of degree n has at most
f(x)  x4  x3  4x2  1 is n  1 turning points.
located. Then draw the graph.
x f(x) Study Tip Example 3 Maximum and Minimum Points
Reading Math Graph f(x)  x3
3x2  5. Estimate the x-coordinates at which the relative

2 9 maxima and relative minima occur.
} change in signs The plurals of maximum

1
1 and minimum are Make a table of values and graph the equation.
} change in signs maxima and minima. f (x )
0 1 f (x )  x 3
3 x 2  5
} change in signs x f (x)
1
3 2 15
2
7 1 1 }
zero between x  2 and x  1
} change in signs 0 5 ← indicates a relative maximum
3 19 1 3 O x
There are zeros between x 
2 2 1 ← indicates a relative minimum
3 5
and x 
1, between x 
1 and
x  0, between x  0 and x  1, 354 Chapter 7 Polynomial Functions
and between x  2 and x  3.
f (x )

O x Unlocking Misconceptions
Modeling Real-World Data Students may incorrectly assume that
functions exactly describe every member of a set of real-world data.
Stress that a function is just a model of the data, and often it is only a
reasonable model for a limited domain of values. Make sure students
f (x)  x 4
x3
4x2  1
understand that the function is just an approximation of the real-world
data and does not completely describe the data.

354 Chapter 7 Polynomial Functions

 Look at the table of values and the graph. MAXIMUM AND
• The values of f(x) change signs between x  2 and x  1, indicating a zero MINIMUM POINTS
of the function.
• The value of f(x) at x  0 is greater than the surrounding points, so it is a In-Class Examples Power
Point®
relative maximum.
• The value of f(x) at x  2 is less than the surrounding points, so it is a 3 Graph f(x)  x3  4x2  5.
relative minimum.
Estimate the x-coordinates at
which the relative maxima
The graph of a polynomial function can reveal trends in real-world data. and relative minima occur.
f (x )
Example 4 Graph a Polynomial Model
ENERGY The average fuel (in gallons) consumed by individual vehicles
f (x)  x3
4x2  5
in the United States from 1960 to 2000 is modeled by the cubic equation
F(t)  0.025t3
1.5t2  18.25t  654, where t is the number of years since 1960.
a. Graph the equation.
O x
Make a table of values for the years 19602000. Plot the points and connect
with a smooth curve. Finding and plotting the points for every fifth year gives
a good approximation of the graph.

t F (t ) F (t )
750
0 654 • The value of f(x) at x  0 is

Consumption (gal)
700
5 710.88 greater than the surrounding
10 711.5 Average Fuel 650 points, so it is a relative
15 674.63 600 maximum.
Energy 20 619 • The value of f(x) at x  3 is
550
Gasoline and diesel fuels 25 563.38 less than the surrounding
are the most familiar
transportation fuels in this 30 526.5 500 points, so it is a relative
country, but other energy 35 527.13 0 10 20 30 40 t minimum.
sources are available, 40 584 Years Since 1960
including ethanol, a grain 4 HEALTH The weight w, in
alcohol that can be
produced from corn or b. Describe the turning points of the graph and its end behavior. pounds, of a patient during a
other crops. There is a relative maximum between 1965 and 1970 and a relative minimum 7-week illness is modeled by
Source: U.S. Environmental
between 1990 and 1995. For the end behavior, as t increases, F(t) increases. the cubic equation w(n) 
Protection Agency
0.1n3  0.6n2  110, where n
c. What trends in fuel consumption does the graph suggest?
is the number of weeks since
Average fuel consumption hit a maximum point around 1970 and then started to the patient became ill.
decline until 1990. Since 1990, fuel consumption has risen and continues to rise.
a. Graph the equation.
w (n)

Patient’s Weight (lb)

A graphing calculator can be helpful in finding the relative maximum and relative
minimum of a function.
110

108
Maximum and Minimum Points
106
You can use a TI-83 Plus to find the coordinates of relative maxima and
relative minima. Enter the polynomial function in the Y list and graph the 0 2 4 6 n
function. Make sure that all the turning points are visible in the viewing Weeks Since Illness Began
window. Find the coordinates of the minimum and maximum points,
respectively. b. Describe the turning points
KEYSTROKES: Refer to page 293 to review finding maxima and minima.
of the graph and its end
(continued on the next page) behavior. There is a relative
www.algebra2.com/extra_examples Lesson 7-2 Graphing Polynomial Functions 355
minimum point at week 4. For
the end behavior, w(n) increases
as n increases.
c. What trends in the patient’s
weight does the graph
Maximum and Minimum Points Remind students of the procedure for find- suggest? The patient lost weight
ing relative minima and maxima using the calculator. First, press 2nd [CALC] and for each of 4 weeks after becom-
select either 3 or 4, depending on whether you are finding a minimum or maxi- ing ill. After 4 weeks, the patient
mum. Then set the left bound. Use the arrow buttons to move the cursor well to gained weight and continues to
the left of the point you suspect is the minimum or maximum, and press ENTER . gain weight.
Move the cursor well to the right of the suspect point. Press ENTER twice to
display the coordinates of the relative maximum/minimum.

Lesson 7-2 Graphing Polynomial Functions 355

 Think and Discuss 1–3. See pp. 407A–407H.

3 Practice/Apply 1. Graph f(x)  x3  3x2  4. Estimate the x-coordinates of the relative

maximum and relative minimum points from the graph.
2. Use the maximum and minimum options from the CALC menu to find the
exact coordinates of these points. You will need to use the arrow keys to
Study Notebook select points to the left and to the right of the point.
1
Have students— 3. Graph f(x) 

x4  4x3  7x2  8. How many relative maximum and
2
relative minimum points does the graph contain? What are the coordinates?
• add the definitions/examples of
the vocabulary terms to their
Vocabulary Builder worksheets for
Chapter 7.
• include any other item(s) that they Concept Check 1. Explain the Location Principle in your own words. See margin.
find helpful in mastering the skills 7. between
2 and 2. State the number of turning points of the graph of a fifth-degree polynomial if it

1, between
1 and has five distinct real zeros. 4
in this lesson. 0, between 0 and 1, 3. OPEN ENDED Sketch a graph of a function that has one relative maximum
and between 1 and 2 point and two relative minimum points. See margin.

Guided Practice Graph each polynomial function by making a table of values. 4–5. See pp. 407A–407H.
About the Exercises… GUIDED PRACTICE KEY 4. f(x)  x3  x2  4x  4 5. f(x)  x4  7x2  x  5
Exercises Examples
Organization by Objective Determine consecutive values of x between which each real zero of each function
• Graph Polynomial 4, 5 1 is located. Then draw the graph. 6–7. See pp. 407A–407H for graphs.
Functions: 13–26
6, 7 2 6. f(x)  x3  x2  1 between
1 and 0 7. f(x)  x4  4x2  2
8, 9 3
• Maximum and Minimum 10
12 4 Graph each polynomial function. Estimate the x-coordinates at which the relative
Points: 13–26 8. Sample answer: maxima and relative minima occur. 8–9. See pp. 407A–407H for graphs.
rel. max. at x 
2, 8. f(x)  x3  2x2  3x  5 9. f(x)  x4  8x2  10
Odd/Even Assignments rel. min. at x  0.5 9. Sample answer: rel. max. at x  0, rel. min. at x 
2 and at x  2
Exercises 13–26 are structured Application CABLE TV For Exercises 10–12, use the following information.
so that students practice the The number of cable TV systems after 1985 can be modeled by the function
same concepts whether they C(t)  43.2t2  1343t  790, where t represents the number of years since 1985.
are assigned odd or even 10. Graph this equation for the years 1985 to 2005. 10–12. See pp. 407A–407H.
problems. 11. Describe the turning points of the graph and its end behavior.
Alert! Exercise 30 involves 12. What trends in cable TV subscriptions does the graph suggest?
research on the Internet or
other reference materials.
Practice and Apply
Assignment Guide Homework Help For Exercises 13–26, complete each of the following.
Basic: 13–25 odd, 27–32, 36–42, For See a. Graph each function by making a table of values.
Exercises Examples
47–66
13–26 1, 2, 3 b. Determine consecutive values of x between which each real zero is located.
Average: 13–25 odd, 27–42, 27–35 4
c. Estimate the x-coordinates at which the relative maxima and relative
47–66 (optional: 43–46) minima occur. 13–26. See pp. 407A–407H.
Extra Practice
Advanced: 14–26 even, 27–60 See page 842. 13. f(x)  x3  4x2 14. f(x)  x3  2x2  6
(optional: 61–66) 15. f(x)  x3  3x2 2 16. f(x)  x3  5x2  9
17. f(x)  3x3  20x2  36x  16 18. f(x)  x3  4x2  2x  1
19. f(x)  x4 8 20. f(x)  x4  10x2  9
Answers 21. f(x)  x4  5x2  2x  1 22. f(x)  x4  x3  8x2  3
1. There must be at least one real 23. f(x)  x4  9x3  25x2  24x  6 24. f(x)  2x4  4x3  2x2  3x  5
zero between two points on a 25. f(x)  x5  4x4  x3  9x2 3 26. f(x)  x5  6x4  4x3  17x2  5x  6
graph when one of the points lies 356 Chapter 7 Polynomial Functions
below the x-axis and the other
point lies above the x-axis.
3. f (x) Differentiated Instruction ELL
Verbal/Linguistic Have the class work in groups of 3 or 4 students.
Instruct students to take turns explaining how to make a table of values
O for a polynomial function, how to plot several points to begin a graph of
x
the function, how to locate the zeros of the function, and how to
estimate the x-coordinates at which the relative maxima and relative
minima of the function occur.

356 Chapter 7 Polynomial Functions

27. highest: 1982; EMPLOYMENT For Exercises 27–30, use the graph that models the Study Guide
NAME ______________________________________________ DATE

andIntervention
Intervention,
____________ PERIOD _____

7-2 Study Guide and

lowest: 2000 unemployment rates from 1975–2000. p. 381 (shown) and p. 382
Graphing Polynomial Functions
28. Rel. max. between 27. In what year was the unemployment Unemployment Graph Polynomial Functions

1980 and 1985 and rate the highest? the lowest? Location Principle
Suppose y  f(x) represents a polynomial function and a and b are two numbers such that
f(a)  0 and f(b)  0. Then the function has at least one real zero between a and b.

Unemployed (Percent
between 1990 and 28. Describe the turning points and end 14

of Labor Force)
Example Determine the values of x between which each real zero of the
1995, rel. min. behavior of the graph. 12 function f(x)  2x4
x3
5 is located. Then draw the graph.
10 Make a table of values. Look at the values of f(x) to locate the zeros. Then use the points to
between 1975 and 29. If this graph was modeled by a 8
sketch a graph of the function.
f (x ) The changes in sign indicate that there are zeros
x f(x)
1980 and between polynomial equation, what is the least 6 2 35
between x  2 and x  1 and between x  1 and
x  2.
1985 and 1990; as the degree the equation could have? 5 4
1
0
2
5
O x

number of years 2 1 4

30. Do you expect the unemployment rate

Lesson 7-2
2 19

increases, the percent 0 5 10 15 20 25

to increase or decrease from 2001 to 2005? Exercises
of the labor force that Years Since 1975
Explain your reasoning. Graph each function by making a table of values. Determine the values of x at
is unemployed which or between which each real zero is located.
1. f(x)  x3  2x2  1 2. f(x)  x4  2x3  5 3. f(x)  x4  2x2  1
decreases.
Online Research
f (x ) f (x ) f (x )

Data Update What is the current unemployment rate?

8
O x O x

30. Sample answer: Visit www.algebra2.com/data_update to learn more.

4

O
–8 –4 4 8x
increase, based on the –4

past fluctuations of CHILD DEVELOPMENT For Exercises 31 and 32, use the following information.
–8

between 0 and
1; between
2 and
3; at 1

the graph The average height (in inches) for boys ages 1 to 20 can be modeled by the
at 1; between 1 and 2 between 1 and 2
4. f(x)  x3  3x2  4 5. f(x)  3x3  2x  1 6. f(x)  x4  3x3  1
equation B(x)  0.001x4  0.04x3  0.56x2  5.5x  25, where x is the age (in f (x ) f (x ) f (x )

years). The average height for girls ages 1 to 20 is modeled by the equation O x
O x

G(x)  0.0002x4  0.006x3  0.14x2  3.7x  26. O x

31. Graph both equations by making a table of values. Use x  {0, 2, 4, 6, 8, 10, 12, at
1, 2 between 0 and 1 between 0 and 1;
14, 16, 18, 20} as the domain. Round values to the nearest inch. See pp. 407A–407H. between 2 and 3

Gl NAME
/M G ______________________________________________
Hill 381 DATE ____________
GlPERIOD
Al _____
b 2
32. Compare the graphs. What do the graphs suggest about the growth rate for both Skills
7-2 Practice,
Practice (Average)
p. 383 and
boys and girls? See margin. Practice, p. 384 Functions
Graphing Polynomial (shown)
Complete each of the following.
a. Graph each function by making a table of values.
PHYSIOLOGY For Exercises 33–35, use the following information. b. Determine consecutive values of x between which each real zero is located.
c. Estimate the x-coordinates at which the relative and relative minima occur.

During a regular respiratory cycle, the volume of air in liters in the human lungs 1. f(x)  x3  3x2  3
f (x )
2. f(x)  x3  1.5x2  6x  1
f (x )

can be described by the function V(t)  0.173t  0.152t2  0.035t3, where t is the
x f(x) x f(x) 8
2 17 2
1 4
1 1 1 4.5
time in seconds. 0
3 O x 0 1 –4 –2 O 2 4x

Child Devolpment
1
5.5 –4

1 1
33. Estimate the real zeros of the function by graphing. 0 and between 5 and 6 2 1 2
9 –8

As children develop, their 3
3 3
3.5

sleeping needs change. 34. About how long does a regular respiratory cycle last? 5 s 4
zeros between
1

19
and 0, 1 and 2,
4
zeros between
2
17
and
1, 0 and 1,
and 2 and 3; rel. max. at x  2, and 3 and 4; rel. max. at x 
1,
Infants sleep about 16–18 35. Estimate the time in seconds from the beginning of this respiratory cycle for the rel. min. at x  0 rel. min. at x  2
3. f(x)  0.75x4  x3  3x2  4 4. f(x)  x4  4x3  6x2  4x  3
hours a day. Toddlers lungs to fill to their maximum volume of air. 3 s x f(x) f (x ) x f(x) f (x )

usually sleep 10–12
3 10.75
3 12

2
4
2
3
hours at night and take
1 O x
1
4 O x

one or two daytime naps. CRITICAL THINKING For Exercises 36–39, sketch a graph of each polynomial. 0
0.75
4 0
3
1 2.75 1 12
School-age children need 36. even-degree polynomial function with one relative maximum and two relative 2 12 2 77
9–11 hours of sleep, and minima 36–39. See pp. 407A–407H for sample graphs. zeros between
3 and
2, and zeros between
3 and
2,

2 and
1; rel. max. at x  0, and 0 and 1; rel. min. at x 
1
teens need at least 9 rel. min. at x 
2 and x  1
hours of sleep. 37. odd-degree polynomial function with one relative maximum and one relative PRICES For Exercises 5 and 6, use the following information.

Source: www.kidshealth.org minimum; the leading coefficient is negative The Consumer Price Index (CPI) gives the relative price

Consumer Price Index

179
for a fixed set of goods and services. The CPI from 178
September, 2000 to July, 2001 is shown in the graph. 177
38. even-degree polynomial function with four relative maxima and three relative Source: U. S. Bureau of Labor Statistics 176
175
5. Describe the turning points of the graph.
minima rel max. in Nov. and June; rel. min in Dec.
174
173
0 1 2 3 4 5 6 7 8 9 10 11
6. If the graph were modeled by a polynomial equation, Months Since September, 2000
39. odd-degree polynomial function with three relative maxima and three relative what is the least degree the equation could have? 4

minima; the leftmost points are negative 7. LABOR A town’s jobless rate can be modeled by (1, 3.3), (2, 4.9), (3, 5.3), (4, 6.4), (5, 4.5),
(6, 5.6), (7, 2.5), (8, 2.7). How many turning points would the graph of a polynomial
function through these points have? Describe them. 4: 2 rel. max. and 2 rel. min.

Gl NAME
/M G ______________________________________________
Hill 384 DATE ____________
Gl PERIOD
Al _____
b 2

40. WRITING IN MATH Answer the question that was posed at the beginning of Reading
7-2 Readingto
to Learn
Learn Mathematics
the lesson. See pp. 407A–407H. Mathematics, p. 385 ELL
Graphing Polynomial Functions
Pre-Activity How can graphs of polynomial functions show trends in data?
How can graphs of polynomial functions show trends in data? Read the introduction to Lesson 7-2 at the top of page 353 in your textbook.
Three points on the graph shown in your textbook are (0, 14), (70, 3.78), and
Include the following in your answer: (100, 9). Give the real-world meaning of the coordinates of these points.
Sample answer: In 1900, 14% of the U. S. population was
foreign born. In 1970, 3.78% of the population was foreign
• a description of the types of data that are best modeled by polynomial born. In 2000, 9% of the population was foreign born.
equations rather than linear equations, and Reading the Lesson
• an explanation of how you would determine when the percent of foreign-born 1. Suppose that f(x) is a third-degree polynomial function and that c and d are real
numbers, with d  c. Indicate whether each statement is true or false. (Remember that
citizens was at its highest and when the percent was at its lowest since 1900. true means always true.)

a. If f(c)  0 and f(d)  0, there is exactly one real zero between c and d. false

www.algebra2.com/self_check_quiz Lesson 7-2 Graphing Polynomial Functions 357 b. If f(c)  f(d)  0, there are no real zeros between c and d. false

Lesson 7-2
c. If f(c)  0 and f(d)  0, there is at least one real zero between c and d. true

2. Match each graph with its description.

a. third-degree polynomial with one relative maximum and one relative minimum;
leading coefficient negative iii
NAME ______________________________________________ DATE ____________ PERIOD _____

Answer Enrichment,
7-2 Enrichment p. 386
b. fourth-degree polynomial with two relative minima and one relative maximum i

c. third-degree polynomial with one relative maximum and one relative minimum;
leading coefficient positive iv

32. The growth rate for both boys and Golden Rectangles d. fourth-degree polynomial with two relative maxima and one relative minimum ii

i. ii. iii. iv.

girls increases steadily until age
f (x ) f (x ) f (x ) f (x )
Use a straightedge, a compass, and the instructions below to construct
a golden rectangle.

18 and then begins to level off, 1. Construct square ABCD with sides of
2 centimeters. Q
D
C
O x O x O x O x

with boys averaging a height of 2. Construct the midpoint of 

midpoint M.
AB. Call the

71 in. and girls a height of 60 in. 3. Using M as the center, set your compass
P
A M
B

opening at MC. Construct an arc with Helping You Remember

B
center M that intersects A. Call the point
of intersection P. 3. The origins of words can help you to remember their meaning and to distinguish
between similar words. Look up maximum and minimum in a dictionary and describe
their origins (original language and meaning). Sample answer: Maximum comes
4. Construct a line through P that is from the Latin word maximus, meaning greatest. Minimum comes from
perpendicular to 
AB. the Latin word minimus, meaning least.

C
5. Extend D so that it intersects the

Lesson 7-2 Graphing Polynomial Functions 357

 Standardized 41. Which of the following could be the graph of f(x)  x3  x2  3x? D
4 Assess Test Practice A f (x ) B f (x )

Open-Ended Assessment
O x O x
Writing Have students write a
paragraph describing how to find
the turning points of the graph of
a polynomial function.
C f (x ) D f (x )

Getting Ready for

Lesson 7-3 O x O x

PREREQUISITE SKILL Lesson 7-3

presents solving equations using
quadratic techniques. Students 42. The function f(x)  x2  4x  3 has a relative minimum located at which of the
will factor polynomials to find the following x values? B
solutions of polynomial equations. A 2 B 2 C 3 D 4
Use Exercises 61–66 to determine
your students’ familiarity with
Graphing Use a graphing calculator to estimate the x-coordinates at which the maxima and

factoring polynomials.
Calculator minima of each function occur. Round to the nearest hundredth.
43. f(x)  x3  x2  7x  3
1.90; 1.23 44. f(x)  x3  6x2  6x  5 3.41; 0.59
45. f(x)  x4  3x2  8 0;
1.22, 1.22 46. f(x)  3x4  7x3  4x  5
Answers 0.52;
0.39, 1.62

53. y Maintain Your Skills

Mixed Review If p(x)  2x2
5x  4 and r(x)  3x3
x2
2, find each value. (Lesson 7-1)
47. r(2a) 24a3
4a2
2 48. 5p(c) 10c2
25c  20 49. p(2a2) 8a 4
10a 2  4
50. r(x  1) 51. p(x2  4) 52. 2[p(x2  1)]  3r(x  1)
3x 3
10x 2  11x
6 2x 4  11x 2  16 4x4
9x 3  28x 2
33x  20
Graph each inequality. (Lesson 6-7) 53
55. See margin.
y  x 2
4x  6
53. y  x2  4x  6 54. y
x2  6x  3 55. y  x2  2x
O x
Solve each matrix equation or system of equations by using inverse matrices.
(Lesson 4-8)
54. y
3 7 1
56. 32 6
1    

a
b

18
(7,
4) 57. 35 4   

m
n

1
(
3,
2)

58. 3j  2k  8 (4,
2) 59. 5y  2z  11 (1, 3)

j  7k  18 10y  4z  2

60. SPORTS Bob and Minya want to build a ramp

O x that they can use while rollerblading. If they want
1
the ramp to have a slope of

, how tall should
4
they make the ramp? (Lesson 2-3) 1 ft
61. (x  5)(x
6)
y 
x 2  6x
3
62. (2b
1)(b
4)
4 ft
55. y Getting Ready for PREREQUISITE SKILL Factor each polynomial.
the Next Lesson (To review factoring polynomials, see Lesson 5-4.)

63. (3a  1)(2a  5) 61. x2  x  30 62. 2b2  9b  4 63. 6a2  17a  5

64. (2m  3)(2m
3) 64. 4m2 9 65. t3  27 66. r4  1
(t
3)(t 2  3t  9) (r 2  1)(r  1)(r
1)
358 Chapter 7 Polynomial Functions

O x
2
y  x
2x

358 Chapter 7 Polynomial Functions

 Graphing
Calculator
A Follow-Up of Lesson 7-2
Investigation
A Follow-Up of Lesson 7-2
Modeling Real-World Data Getting Started
You can use a TI-83 Plus to model data whose curve of best fit is a polynomial function.

Knowing Your Calculator

Example Students should clear lists L1 and
The table shows the distance a seismic wave can travel based on its distance from
an earthquake’s epicenter. Draw a scatter plot and a curve of best fit that relates L2 before entering the data from
distance to travel time. Then determine approximately how far from the epicenter the table in Step 1. This is a more
the wave will be felt 8.5 minutes after the earthquake occurs. reliable approach than simply
Source: University of Arizona “overwriting” old data with new
Travel Time (min) 1 2 5 7 10 12 13 data.
Distance (km) 400 800 2500 3900 6250 8400 10,000

Enter the travel times in L1 and the

distances in L2.
Graph the scatter plot. Teach
KEYSTROKES: Refer to page 87 to review
KEYSTROKES: Refer to page 87 to review how to graph a scatter plot. • Step 3 asserts that a quartic
how to enter lists. curve will fit the data best. An
easy way to verify this is to find
Compute and graph the equation for Use the [CALC] feature to find the
the curve of best fit. A quartic curve is value of the function for x  8.5. quadratic and cubic regression
the best fit for these data. KEYSTROKES: Refer to page 87 to review
equations and then copy them
KEYSTROKES: STAT 7 2nd how to find function values. to the Y= list along with the
, quartic regression equation.
[L1] 2nd [L2] ENTER
Change the window settings for
VARS 5 1 GRAPH
the x-axis to [0, 18.8]. Turn off
the scatter plot, and graph all
three regression equations on
the same screen. Then use the
[0, 15] scl: 1 by [0, 10000] scl: 500 Trace feature to go to each x
value in the first row of the
After 8.5 minutes, you would expect the wave to be felt approximately 5000 kilometers away.
table. While at each x value, use
the up/down arrow keys to
Exercises 1. See pp. 407A–407H. move between the curves,
Year Minutes
Use the table that shows how many minutes out of each eight-hour
comparing the y value for each
1940 83
work day are used to pay one day’s worth of taxes. regression curve with the
1950 117
1. Draw a scatter plot of the data. Then graph several curves of best fit that y value given in the table.
1960 130
relate the number of minutes to the year. Try LinReg, QuadReg, and CubicReg. • Have students complete
1970 141
2. Write the equation for the curve that best fits the data. See margin. 1980 145 Exercises 1–3.
3. Based on this equation, how many minutes should you expect to work 1990 145
each day in the year 2010 to pay one day’s taxes? about 184 min 2000 160
Source: Tax Foundation Assess
www.algebra2.com/other_calculator_keystrokes
In Exercises 1 and 2, make sure
Investigating Slope-Intercept Form 359 students have graphed several
Graphing Calculator Investigation Modeling Real-World Data 359
curves before choosing the one
they feel best fits the data.
Students should be able to justify
Answer their final choice.
2. (9.4444444  10
4)x 3
0.1057143x 2  4.21031746x
83.1904762; In Exercise 3, students’ answers
The equation is a good fit because r 2  0.994. may vary slightly depending on
their best-fit curve.

Graphing Calculator Investigation Modeling Real-World Data 359

 Lesson Solving Equations Using
Notes Quadratic Techniques
• Write expressions in quadratic form.

1 Focus • Use quadratic techniques to solve equations.

can solving polynomial equations

Vocabulary help you to find dimensions?
5-Minute Check • quadratic form 50
2x
Transparency 7-3 Use as a The Taylor Manufacturing Company makes x x
quiz or review of Lesson 7-2. open metal boxes of various sizes. Each sheet x x
of metal is 50 inches long and 32 inches wide. 32
2x
Mathematical Background notes To make a box, a square is cut from each corner. x
x x
x
The volume of the box depends on the side
are available for this lesson on length x of the cut squares. It is given by
p. 344C. V(x)  4x3  164x2  1600x. You can solve a polynomial equation to find
the dimensions of the square to cut for a box with specific volume.
Building on Prior
Knowledge QUADRATIC FORM In some cases, you can rewrite a polynomial in x in the
form au2  bu  c. For example, by letting u  x2 the expression x4  16x2  60 can
In Chapter 6, students learned to TEACHING TIP be written as (x2)2  16(x2)  60 or u2  16u  60. This new, but equivalent,
solve quadratic equations. This method of expression is said to be in quadratic form .
Students should recognize that substituting u for x 2 is
the same techniques are used in called u substitution.
Quadratic Form
this lesson to solve higher-degree
polynomial equations that can be An expression that is quadratic in form can be written as au2  bu  c for any
numbers a, b, and c, a  0, where u is some expression in x. The expression
written using quadratic form. au2  bu  c is called the quadratic form of the original expression.

can solving polynomial

equations help you to Example 1 Write an Expression in Quadratic Form
find dimensions?
Write each expression in quadratic form, if possible.
Ask students: a. x4  13x2  36
• How does this metal sheet x4  13x2  36  (x2)2  13(x2)  36 (x2)2 = x4
become a box? by folding up the
sides after the squares have been b. 16x6
625
removed from the corners 16x6  625  (4x3)2  625 (x3)2 = x6

• Why are squares, and not c. 12x8
x2  10

rectangles, cut from the corners? This cannot be written in quadratic form since x8  (x2)2.
Squares are used so that the sides

1

of the box have the same height. If d. x
9x 2  8
x  9x 2  8  x 2   9x 2   8 x1 = x
2


1

1
2
1

rectangles are cut from the corners, 1 2

there are two different heights for the

sides and a box cannot be formed.
SOLVE EQUATIONS USING QUADRATIC FORM In Chapter 6, you
• What are the length, width, learned to solve quadratic equations by using the Zero Product Property and the
and height of the box in terms Quadratic Formula. You can extend these techniques to solve higher-degree
of x? length: 50
2x; width: polynomial equations that can be written using quadratic form or have an
32
2x; height: x expression that contains a quadratic factor.
360 Chapter 7 Polynomial Functions

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 7 Resource Masters 5-Minute Check Transparency 7-3
• Study Guide and Intervention, pp. 387–388 Answer Key Transparencies
• Skills Practice, p. 389
• Practice, p. 390 Technology
• Reading to Learn Mathematics, p. 391 Alge2PASS: Tutorial Plus, Lesson 13
• Enrichment, p. 392 Interactive Chalkboard
• Assessment, p. 443
 Example 2 Solve Polynomial Equations
Solve each equation. 2 Teach
a. x4
13x2  36  0
x4  13x2  36  0 Original equation
QUADRATIC FORM
(x2)2  13(x2)  36  0 Write the expression on the left in quadratic form.
In-Class Example Power
Point®
(x2  9)(x2  4)  0 Factor the trinomial.
(x  3)(x  3)(x  2)(x  2)  0 Factor each difference of squares.
1 Write each expression in
Use the Zero Product Property. quadratic form, if possible.
x  3  0 or x  3  0 or x20 or x20
a. 2x6  x3  9 2(x 3)2  (x 3)  9
x3 x  3 x2 x  2
The solutions are 3, 2, 2, and 3.
b. 7x10  6 7(x 5)2  6
c. x4  2x3  1 This cannot be
CHECK The graph of f (x )
f(x)  x4  13x2  36 written in quadratic form since
40
shows that the graph x 4  (x 3)2.
intersects the x-axis 2
d. x 3  2x 3  4 (x 3 )  2(x 3 )
4
20 2 1 1 1




 
at 3, 2, 2, and 3. ⻫

2 O 2 x
Study Tip f (x )  x 4
13x 2  36

Look Back
To review the formula for b. x3  343  0 SOLVE EQUATIONS USING
factoring the sum of two
cubes, see Lesson 5-4.
x3  343  0 Original equation QUADRATIC FORM
(x)3  73  0 This is the sum of two cubes.
(x  7)[x  x(7)  72]  0
2 Sum of two cubes formula with a = x and b = 7 In-Class Examples Power
Point®
(x  7)(x2  7x  49)  0 Simplify.
x  7  0 or x2  7x  49  0 Zero Product Property 2 Solve each equation.
The solution of the first equation is 7. The second equation can be solved by a. x4  29x2  100  0
using the Quadratic Formula.
5,
2, 2, 5
b   
b  4ac 2
b. x3  216  0
6, 3  3i
3,
x 

2a
Quadratic Formula
3
3i
3
(7)  
(7)  
 4(1)(49) 2




Replace a with 1, b with 7, and c with 49. 1 1
2(1) 3 Solve x
2
 x
4
 6  0. 81




7  147
Simplify.
2
 or

7  i147 7  7i3



2


2   1
147  = i147


Study Tip 7  7i3 7  7i3

Thus, the solutions of the original equation are 7,

, and

.
2 2
Substitution
To avoid confusion, you
can substitute another Some equations involving rational exponents can be solved by using a quadratic
variable for the expression technique.
in parentheses.
For example, Example 3 Solve Equations with Rational Exponents
x
3
2  6x
3
  5  0
1 1

2

1

could be written as Solve x 3
6x 3  5  0.
u2  6u  5  0. Then,
2

1

once you have solved the x 3  6x 3  5  0 Original equation
equation for u, substitute
x
3
2  6x
3
  5  0
1 1

1

x 3 for u and solve for x. Write the expression on the left in quadratic form.
(continued on the next page)
www.algebra2.com/extra_examples Lesson 7-3 Solving Equations Using Quadratic Techniques 361

Unlocking Misconceptions
Quadratic Form In Example 1 on p. 360, students may mistakenly con-
clude that variables must have even powers in order for the expression
to be written in quadratic form. Draw students’ attention to Example 1d.
Clarify that the relationship between the powers of two terms is what
indicates whether an expression can be written in quadratic form. In
1



Example 1d, the power of the x term is twice the power of the x 2 term,
so the expression can be rewritten in quadratic form.

Lesson 7-3 Solving Equations Using Quadratic Techniques 361

 x
3
 1x
3
 5  0
1 1
In-Class Example Power
Point®
Factor the trinomial.

1

1

x 10 x 50
4 Solve x  x  12. 9 or Zero Product Property
3 3

1

1

x 1 3 x 5 3 Isolate x on one side of the equation.

x   13

1

3
3
x   53

1

3
3
Cube each side.

3 Practice/Apply x1 x  125 Simplify.

CHECK Substitute each value into the original equation.

2

1

2

1

x 3  6x 3  5  0 x 3  6x 3  5  0
Study Notebook
2

1

2

1

1 3  6(1) 3  5
0 125 3  6(125) 3  5
0
Have students—
• add the definitions/examples of 165
0 25  30  5
0

the vocabulary terms to their 00 ⻫ 00 ⻫

Vocabulary Builder worksheets for The solutions are 1 and 125.

Chapter 7.
• include any other item(s) that they To use a quadratic technique, rewrite the equation so one side is equal to zero.
find helpful in mastering the skills
Example 41 Solve Radical Equations
in this lesson.
Solve x
6
x  7.
x  6x  7 Original equation
x  6x  7  0 Rewrite so that one side is zero.
x 2  6x   7  0 Write the expression on the left in quadratic form.
About the Exercises… You can use the Quadratic Formula to solve this equation.
Organization by Objective b  
b2  4
ac
• Quadratic Form: 11–16 x 

Quadratic Formula
2a
• Solve Equations Using

(6)  (6) 2
 4(1)
Quadratic Form: 17–30 x 




2(1)
(7)
Replace a with 1, b with 6, and c with 7.

Odd/Even Assignments 68

x 
2

Simplify.
Exercises 11–30 are structured
68 68
so that students practice the x 
2



or x
2

Write as two equations.

same concepts whether they

x  7 x  1 Simplify.
are assigned odd or even
problems. Study Tip x  49
Look Back Since the principal square root of a number cannot be negative, the equation
Assignment Guide To review principal roots, x  1 has no solution. Thus, the only solution of the original equation is 49.
see Lesson 5-5.
Basic: 11–27 odd, 32–34, 37–52
Average: 11–31 odd, 32–34,
37–52
Advanced: 12–30 even, 32–48
(optional: 49–52) Concept Check 1. OPEN ENDED Give an example of an equation that is not quadratic but can be
written in quadratic form. Then write it in quadratic form. 1–3. See margin.
All: Practice Quiz 1 (1–5) 2. Explain how the graph of the related polynomial function can help you verify the
solution to a polynomial equation.
3. Describe how to solve x5  2x3  x  0.
Answers
362 Chapter 7 Polynomial Functions
1. Sample answer: 16x 4
12x 2  0;
4[4(x 2)2
3x 2]  0
2. The solutions of a polynomial
equation are the points at which
Differentiated Instruction
the graph intersects the x-axis. Visual/Spatial Have students repeat Example 3, using the substitution
3. Factor out an x and write the method discussed in the Study Tip at the bottom of p. 361. Instruct stu-
equation in quadratic form so you dents to write the given equation in pencil and then use a colored pencil to
1



have x[(x 2)2
2(x 2)  1]  0. write the statement “Let u  x .” Have students continue solving the prob-
3

Factor the trinomial and solve for lem using the colored pencil to help them remember they are working
1

x using the Zero Product Property. with a substituted variable. After solving for u, when students substitute x 3
The solutions are
1, 0, and 1. for u they should resume using their regular pencil.

362 Chapter 7 Polynomial Functions

 Guided Practice Write each expression in quadratic form, if possible. Study
7-3 Guide
Study
NAME ______________________________________________ DATE

andIntervention
Guide and Intervention,
____________ PERIOD _____

GUIDED PRACTICE KEY 4. 5y4  7y3  8 not possible 5. 84n4  62n2 84(n2)2
62(n2) p. 387
Solving(shown) and
Equations Using p. 388
Quadratic Techniques
Quadratic Form Certain polynomial expressions in x can be written in the quadratic
Exercises Examples form au2  bu  c for any numbers a, b, and c, a  0, where u is an expression in x.

4, 5 1
,
3
3i
3
Solve each equation. 8. 6,
3  3i
3 Example Write each polynomial in quadratic form, if possible.

6–9 2 6. x3  9x2  20x  0 0,
5,
4 7. x4  17x2  16  0
4,
1, 4, 1 a. 3a6
9a3  12

Let u  a3.
10 3
1

3a6  9a3  12  3(a3)2  9(a3)  12

8. x3  216  0 9. x  16x 2  64 64 b. 101b
49
b   42

Let u  b
.
101b  49b  )2  49(b
  42  101(b  )  42

Application 10. POOL The Shelby University swimming pool is in the shape of a rectangular
c. 24a5  12a3  18
This expression cannot be written in quadratic form, since a5  (a3)2.

prism and has a volume of 28,000 cubic feet. The dimensions of the pool are x Exercises
feet deep by 7x  6 feet wide by 9x  2 feet long. How deep is the pool? 8 ft Write each polynomial in quadratic form, if possible.

1. x4  6x2  8 2. 4p4  6p2  8

★ indicates increased difficulty 11. 2(x 2)2  6(x 2)
10 12. not possible 13. 11(n 3)2  44(n 3) 15. not possible (x 2) 2  6(x 2)
8 4(p2)2  6(p 2)  8
1



1



3. x8  2x4  1 4. x 8  2x 16  1

Practice and Apply (x 4)2  2(x 4)  1 x16 

1 2
 2x 16   1
1


Lesson 7-3
5. 6x4  3x3  18 6. 12x4  10x2  4
12(x 2)2  10(x 2)
4
Write each expression in quadratic form, if possible. 14. b[7(b 2)2
4(b 2)  2)]
not possible
Homework Help 7. 24x8  x4  4 8. 18x6  2x3  12
For See
Exercises Examples 11. 2x4  6x2  10 12. a8  10a2  16 13. 11n6  44n3 24(x 4)2  x 4  4 18(x 3)2
2(x 3)  12

1

1

2

1
9. 100x4  9x2  15 10. 25x8  36x6  49

11–16 1 14. 7b5  4b3  2b 15. 7x  3x  4
9 3 16. 6x  4x  16 5 5
100(x 2)2
9(x 2)
15 not possible

6x 5 
4x 5 
16  0
17–28 2–4

2 1


1
11. 48x6  32x3  20 12. 63x8  5x4  29
29–36 2 48(x3)2
32(x 3)  20 63(x 4)2  5(x 4)
29
Solve each equation. 17–28. See pp. 407A–407H. 13. 32x10  14x5  143 14. 50x3  15xx  18
2
50x 2 
15x 2 
18
3 3

Extra Practice 17. m4  7m3  12m2  0 18. a5  6a4  5a3  0 19. b4  9

 
32(x 5)2  14(x 5)
143
60x6  7x3 3 10x10  7x5 7
See page 842. 15. 16.
20. t5  256t  0 21. d4  32  12d2 22. x4  18  11x2 60(x 3)2
7(x 3)  3 10(x 5)2
7(x 5)
7

1

1

23. x3  729  0 24. y3  512  0 25. x  8x  15  0
2 4 Gl NAME
/M G ______________________________________________
Hill 387 DATE ____________
GlPERIOD
Al _____
b 2

Skills
7-3 Practice,
Practice p. 389 and

2

1
(Average)
26. p  11p  28  0
3 3 27. y  19y  60 28. z  8z  240 Practice, p. 390
Solving Equations (shown)
Using Quadratic Techniques

★ 29. s3  4s2  s  4  0 1,
1,
4 ★ 30. h3  8h2  3h  24  0 8

Write each expression in quadratic form, if possible.

1. 10b4  3b2  11 2. 5x8  x2  6 3. 28d6  25d3

10(b 2)2  3(b 2)
11 not possible 28(d3)2  25(d3)

31. GEOMETRY The width of a rectangular prism is w centimeters. The height is 4. 4s8  4s4  7 5. 500x4  x2 6. 8b5  8b3  1

4(s4)2  4(s4)  7 500(x2)2
x2

2 centimeters less than the width. The length is 4 centimeters more than the 2



1



not possible
1



1



7. 32w5  56w3  8w 8. e 3  7e 3  10 9. x 5  29x 10  2
width. If the volume of the prism is 8 times the measure of the length, find the (e ) 1 1 1
 7(e
1
8w[4(w 2)2
7(w 2)  1]
 2 
)
10 (x )2  29(x )  2
 

dimensions of the prism. w  4 cm,   8 cm, h  2 cm

3 3 10 10

Solve each equation.

10. y4  7y3  18y2  0
2, 0, 9 11. s5  4s4  32s3  0
8, 0, 4

DESIGN For Exercises 32
34, use the following information.
12. m4  625  0
5, 5,
5i, 5i 13. n4  49n2  0 0,
7, 7
Jill is designing a picture frame for an art project. She
14. x4  50x2  49  0
1, 1,
7, 7 15. t4  21t2  80  0
4, 4,
5
,

5

plans to have a square piece of glass in the center and
3 3
surround it with a decorated ceramic frame, which will 16. 4r6  9r4  0 0, ,
 17. x4  24  2x2
2, 2,
i
6
, i
6

2 2

also be a square. The dimensions of the glass and frame 18. d 4  16d 2  48
2, 2,
2
3
, 2
3

7
7i
3
19. t3  343  0 7,   , 

7  7i
3

2 2
are shown in the diagram at the right. Jill determines that 1



1



20. x  5x  6  0 16, 81
2 4
4



2



21. x 3  29x 3  100  0 8, 125
x
Designer she needs 27 square inches of material for the frame. 3



22. y3  28y  27  0 1, 9
2
23. n  10n
  25  0 25

Designers combine practical 32. Write a polynomial equation that models the area of 24. w  12w
  27  0 9, 81 25. x  2x  80  0 100

knowledge with artistic the frame. x 4
7x 2  9  27

ability to turn abstract ideas 26. PHYSICS A proton in a magnetic field follows a path on a coordinate grid modeled by
33. What are the dimensions of the glass piece? 3 in.  3 in. the function f(x)  x4  2x2  15. What are the x-coordinates of the points on the grid
where the proton crosses the x-axis?

5 ,
5
into formal designs. x 2
3 in.
Designers usually specialize 34. What are the dimensions of the frame? 6 in.  6 in. 27. SURVEYING Vista county is setting aside a large parcel of land to preserve it as open
space. The county has hired Meghan’s surveying firm to survey the parcel, which is in
in a particular area, such as the shape of a right triangle. The longer leg of the triangle measures 5 miles less than
the square of the shorter leg, and the hypotenuse of the triangle measures 13 miles less
clothing, or home interiors. than twice the square of the shorter leg. The length of each boundary is a whole number.
PACKAGING For Exercises 35 and 36, use the following information. Find the length of each boundary. 3 mi, 4 mi, 5 mi

A computer manufacturer needs to change the dimensions of its foam packaging for NAME ______________________________________________
390 DATE ____________
Gl PERIOD
Al _____
Online Research a new model of computer. The width of the original piece is three times the height,
Gl
Reading
7-3
/M

Readingto
G Hill

to Learn
Learn Mathematics
b 2

For information about Mathematics, p. 391 ELL

and the length is equal to the height squared. The volume of the new piece can be Solving Equations Using Quadratic Techniques
a career as a designer,
represented by the equation V(h)  3h4  11h3  18h2  44h  24, where h is the Pre-Activity How can solving polynomial equations help you to find dimensions?

visit: Read the introduction to Lesson 7-3 at the top of page 360 in your textbook.
height of the original piece. 35. h2  4, 3h  2, h  3 Explain how the formula given for the volume of the box can be obtained
www.algebra2.com/ from the dimensions shown in the figure.

careers ★ 35. Factor the equation for the volume of the new piece to determine three Sample answer: The volume of a rectangular box is given
by the formula V 
wh. Substitute 50
2x for
,
32
2x for w, and x for h to get
expressions that represent the height, length, and width of the new piece. V(x)  (50
2x)(32
2x)(x)  4x 3
164x 2  1600x.

★ 36. How much did each dimension of the packaging increase for the new foam Reading the Lesson
piece? The height increased by 3, the width increased by 2, and the length 1. Which of the following expressions can be written in quadratic form? b, c, d, f, g, h, i
increased by 4. a. x3  6x2  9 b. x4  7x2  6 c. m6  4m3  4
www.algebra2.com/self_check_quiz Lesson 7-3 Solving Equations Using Quadratic Techniques 363 d. y  2y 2  15
1



e. x5  x3  1 f. r4  6  r8
1 1 1 1









g. p 4  8p 2  12 h. r 3  2r 6  3 i. 5z  2z  3

2. Match each expression from the list on the left with its factorization from the list on
NAME ______________________________________________ DATE ____________ PERIOD _____ the right.

a. x4  3x2  40 vi i. (x3  3)(x3  3)

Enrichment, p. 392 Lesson 7-3
7-3 Enrichment
b. x4  10x2  25 v ii. (x  3)(x  3)

c. x6  9 i iii. (x  3)2

Odd and Even Polynomial Functions d. x  9 ii iv. (x2  1)(x4  x2  1)
Functions whose graphs are symmetric with Functions whose graphs are symmetric with
respect to the origin are called odd functions. respect to the y-axis are called even functions. e. x6  1 iv v. (x2  5)2
If f(x)  f(x) for all x in the domain of f(x), If f (x)  f(x) for all x in the domain of f(x), f. x  6x  9 iii vi. (x2  5)(x2  8)
then f (x) is odd. then f (x) is even.
f (x ) f (x )
4
6
2 Helping You Remember
f (x )  1–2x 3 4
3. What is an easy way to tell whether a trinomial in one variable containing one constant
–2 –1 O 1 2x
2 term can be written in quadratic form?
–2 f (x )  1–4x 4  4

–4
–2 –1 O 1 2x Sample answer: Look at the two terms that are not constants and
compare the exponents on the variable. If one of the exponents is twice
the other, the trinomial can be written in quadratic form.
Example Determine whether f(x)  x3
3x is odd, even, or neither.
f(x)  x3  3x
f(x)  (x)3  3(x) Replace x with x. f (x )
4
 x3  3x Simplify. f (x ) x3 3x

Lesson 7-3 Solving Equations Using Quadratic Techniques 363

 37. Write the equation 37. CRITICAL THINKING Explain how you would solve a  32  9a  3 8.
4 Assess in quadratic form,
u 2
9n  8  0,
Then solve the equation.

38. WRITING IN MATH Answer the question that was posed at the beginning of
where u  a
3.
Open-Ended Assessment Then factor and use the lesson. See pp. 407A–407H.
the Zero Product How can solving polynomial equations help you to find dimensions?
Speaking Have students explain Property to solve for Include the following items in your answer:
how the equation in Example 4 on a; 11, 4, 2, and
5. • an explanation of how you could determine the dimensions of the cut square
p. 362 can be solved by first sub- if the desired volume was 3600 cubic inches, and
stituting a for x. If students have • an explanation of why there can be more than one square that can be cut to
difficulty getting started, ask them produce the same volume.
how they could express x in
terms of a, given that a  x. Standardized 39. Which of the following is a solution of x4  2x2  3  0? D
Test Practice A 2
4
B 1 C 3 D 3
Getting Ready for 40. EXTENDED RESPONSE Solve 18x  92x   4  0 by first rewriting it in
Lesson 7-4 quadratic form. Show your work.
1

18
PREREQUISITE SKILL Lesson 7-4
introduces students to the
Maintain Your Skills
Remainder and Factor Theorems. Mixed Review Graph each function by making a table of values. (Lesson 7-2)
Students will use division to find 41–42. See 41. f(x)  x3  4x2  x  5 42. f(x)  x4  6x3  10x2  x  3
the factors of polynomials. Use pp. 407A–407H.
Exercises 49–52 to determine your Find p(7) and p(
3) for each function. (Lesson 7-1)
2
students’ familiarity with dividing 43. p(x)  x2  5x  3 44. p(x)  x3  11x  4
45. p(x) 

x4  3x3
3
17; 27 262; 2 1715
polynomials by a binomial.

; 135
For Exercises 46–48, use the following information. 3
Triangle ABC with vertices A(2, 1), B( 3, 3), and C( 3, 1) is rotated 90°
Assessment Options counterclockwise about the origin. (Lesson 4-4)

Practice Quiz 1 The quiz 46. Write the coordinates of the triangle in a vertex matrix.

3

21

3
1
3

provides students with a brief 47. Find the coordinates of the ABC. A (
1,
2), B (3,
3), C (1, 3)
review of the concepts and skills 48. Graph the preimage and the image. See margin.
in Lessons 7-1 through 7-3.
Lesson numbers are given to the
Getting Ready for PREREQUISITE SKILL Find each quotient.
the Next Lesson (To review dividing polynomials, see Lesson 5-3.) 49–52. See margin.
right of the exercises or 49. (x3  4x2  9x  4)  (x  1) 50. (4x3  8x2  5x  10)  (x  2)
instruction lines so students can
51. (x4  9x2  2x  6)  (x  3) 52. (x4  3x3  8x2  5x  6)  (x  1)
review concepts not yet
mastered.
Quiz (Lessons 7-1 through 7-3)
is available on p. 443 of the P ractice Quiz 1 Lessons 7-1 through 7-3
Chapter 7 Resource Masters.
1. If p(x)  2x3  x, find p(a  1). (Lesson 7-1) 2a 3
6a2  5a
1
2. Describe the end behavior of the graph at the right. Then determine whether f (x )
8
Answers it represents an odd-degree or an even-degree polynomial function and state
the number of real zeros. (Lesson 7-1) See margin. 4
48. y
3. Graph y  x3  2x2  4x  6. Estimate the x-coordinates at which the
C' relative maxima and relative minima occur. (Lesson 7-2) See pp. 407A–407H.
4
2 O 2 4x

1

2

4
A 4. Write the expression 18x  36x  5 in quadratic form. (Lesson 7-3)
3 3

See margin.
8
, i
3
5. Solve a4  6a2  27. (Lesson 7-3)
3, 3,
i
3
O C x
A'
364 Chapter 7 Polynomial Functions
B B'

49. x 2  5x
4 Answers (Practice Quiz 1)

64
50. 4x 2
16x  27
 2. f(x) →
 as x → , f(x) →  as x →
; odd; 3
x2
( )2  3(6x )  5 or 36(
3 x)2  18(
3 x)  5
1 1
 
54 4. 6x 3 3

51. x3
6x
20

x
3

21
52. x3  2x2
10x  15

x1

364 Chapter 7 Polynomial Functions

 The Remainder and Lesson
Factor Theorems Notes

• Evaluate functions using synthetic substitution.

• Determine whether a binomial is a factor of a polynomial by using synthetic
substitution. 1 Focus
Vocabulary can you use the Remainder
• synthetic substitution 5-Minute Check
Theorem to evaluate polynomials?
• depressed polynomial Transparency 7-4 Use as a
The number of international travelers to the United quiz or review of Lesson 7-3.
States since 1986 can be modeled by the equation
T(x)  0.02x3  0.6x2  6x  25.9, where x is the Mathematical Background notes
number of years since 1986 and T(x) is the number
of travelers in millions. To estimate the number of
are available for this lesson on
travelers in 2006, you can evaluate the function for p. 344D.
x  20, or you can use synthetic substitution.
can you use the
Remainder Theorem to
Study Tip SYNTHETIC SUBSTITUTION Synthetic division is a shorthand method of evaluate polynomials?
Look Back long division. It can also be used to find the value of a function. Consider the
To review dividing polynomial function f(a)  4a2  3a  6. Divide the polynomial by a  2. Ask students:
polynomials and • What would be the value of x if
synthetic division, Method 1 Long Division Method 2 Synthetic Division
see Lesson 5-3. you wanted to use the function
4a  5 2 4 3 6
to estimate the number of
a  2 4
a2
3a
6 8 10
4a2  8a
travelers in 1987? 1
4 5 16
5a  6 • Would you expect the actual
5a  10 number of travelers in 2006 to
16 exactly match the number
Compare the remainder of 16 to f(2). predicted by the function?
f(2)  4(2)2  3(2)  6 Replace a with 2.
Explain. No. Sample answer: The
equation is a model based on data
 16  6  6 Multiply.
available at this time.
 16 Simplify.
• Do you think the model is more
Notice that the value of f(2) is the same as the remainder when the polynomial is accurate for the years immedi-
divided by a  2. This illustrates the Remainder Theorem .
ately following 1986 or for years
in the future? Explain. Sample
Remainder Theorem answer: The model closely matches
If a polynomial f(x) is divided by x – a, the remainder is the constant actual data for the years immedi-
f(a), and ately following 1986. For years in
Dividend equals quotient times divisor plus remainder. the future, the model is less likely











f(x)  q(x) (x – a)  f(a), to be as accurate.

where q(x) is a polynomial with degree one less than the degree of f(x).

When synthetic division is used to evaluate a function, it is called synthetic

substitution . It is a convenient way of finding the value of a function, especially
when the degree of the polynomial is greater than 2.
Lesson 7-4 The Remainder and Factor Theorems 365

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 7 Resource Masters Teaching Algebra With Manipulatives 5-Minute Check Transparency 7-4
• Study Guide and Intervention, pp. 393–394 Masters, p. 252 Answer Key Transparencies
• Skills Practice, p. 395
• Practice, p. 396 Technology
• Reading to Learn Mathematics, p. 397 Interactive Chalkboard
• Enrichment, p. 398

Lesson x-x Lesson Title 365

Building on Prior Example 1 Synthetic Substitution
Knowledge If f(x)  2x4
5x2  8x
7, find f(6).
In Lesson 5-3, students learned Method 1 Synthetic Substitution
synthetic division. In this lesson, By the Remainder Theorem, f(6) should be the remainder when you divide the
students will use synthetic divi- polynomial by x  6.
sion to evaluate a function and 6 2 0 5 8 7 Notice that there is no x3 term. A zero
to find factors of polynomials. 12 72 402 2460 is placed in this position as a placeholder.
2 12 67 410 2453
The remainder is 2453. Thus, by using synthetic substitution, f(6)  2453.

2 Teach Method 2 Direct Substitution

Replace x with 6.
SYNTHETIC SUBSTITUTION
f(x)  2x4  5x2  8x  7 Original function

In-Class Example Power

Point®
f(6)  2(6)4  5(6)2  8(6)  7 Replace x with 6.

 2592  180  48  7 or 2453 Simplify.

Teaching Tip Most of this By using direct substitution, f(6)  2453.
lesson relies heavily on synthetic
substitution. Before finishing your
discussion of Example 1, be
certain that students understand FACTORS OF POLYNOMIALS Divide f(x)  x4  x3  17x2  20x  32
the method. Ask a student volun- by x  4.
teer to demonstrate the steps of 4 1 1 17 20 32
the synthetic substitution shown 4 20 12 32
in Method 1 of Example 1, and Study Tip 1 5 3 8 0
invite students to discuss any
problems they have with the Depressed The quotient of f(x) and x  4 is x3  5x2  3x  8. When you divide a polynomial
Polynomial by one of its binomial factors, the quotient is called a depressed polynomial . From
technique before moving on. A depressed polynomial
the results of the division and by using the Remainder Theorem, we can make the
has a degree that is one
1 If f(x)  3x4  2x3  x2  2, less than the original
following statement.
polynomial.
find f(4). 654 Dividend equals quotient times divisor plus remainder.









x4  x3  17x2  20x  32  (x3  5x2  3x  8) (x  4)  0

Since the remainder is 0, f(4)  0. This means that x  4 is a factor of

x4  x3  17x2  20x  32. This illustrates the Factor Theorem , which is a
special case of the Remainder Theorem.

Factor Theorem
The binomial x  a is a factor of the polynomial f(x) if and only if f(a)  0.

Suppose you wanted to find the factors of f (x )

16
x3  3x2  6x  8. One approach is to graph
the related function, f(x)  x3  3x2  6x  8. 8
From the graph at the right, you can see that
the graph of f(x) crosses the x-axis at 2, 1, and 4.
2 O 2 4 x
These are the zeros of the function. Using these

8
zeros and the Zero Product Property, we can
express the polynomial in factored form.
16 f (x )  x 3
3 x 2
6 x  8

366 Chapter 7 Polynomial Functions

366 Chapter 7 Polynomial Functions

 f(x)  [x  (2)](x  1)(x  4) FACTORS OF
 (x  2)(x  1)(x  4) POLYNOMIALS
This method of factoring a polynomial has its limitations. Most polynomial
functions are not easily graphed and once graphed, the exact zeros are often Teaching Tip Remind students that
difficult to determine.
not all polynomials can be factored.
The Factor Theorem can help you find all factors of a polynomial. Emphasize that the factors indicate
where the graph of the function
crosses the x-axis. If the graph of a
Example 2 Use the Factor Theorem polynomial function has no x-inter-
Show that x  3 is a factor of x3  6x2
x
30. Then find the remaining factors cepts, then the polynomial cannot
of the polynomial. be factored. Students can graph the
The binomial x  3 is a factor of the polynomial if 3 is a zero of the related function f(x)  x4  x3  x2  2 to
polynomial function. Use the Factor Theorem and synthetic division. see an example of a polynomial
3 1 6 1 30 that cannot be factored.
3 9 30
1 3 10 0
Study Tip In-Class Examples Power
Point®
Since the remainder is 0, x  3 is a factor of the polynomial. The polynomial
Factoring x3  6x2  x  30 can be factored as (x  3)(x2  3x  10). The polynomial
The factors of a x2  3x  10 is the depressed polynomial. Check to see if this polynomial 2 Show that x  3 is a factor of
polynomial do not have can be factored. x3  4x2  15x  18. Then
to be binomials. For
x2  3x  10  (x  2)(x  5) Factor the trinomial. find the remaining factors of
example, the factors of
x3  x2  x  15 are
the polynomial.
So, x3  6x2  x  30  (x  3)(x  2)(x  5).
x  3 and x2  2x  5.
3 1 4
15
18
CHECK You can see that the graph of the related 3 21 18
function f(x)  x3  6x2  x  30 crosses
the x-axis at 5, 3, and 2. Thus, 1 7 6 0
f(x)  [x  (5)][x  (3)](x  2). ⻫
So, x 3  4x 2
15x
18 
(x
3)(x 2  7x  6). Since
x 2  7x  6  (x  1)(x  6),
x 3  4x 2
15x
18 
Example 3 Find All Factors of a Polynomial (x
3)(x  1)(x  6).
GEOMETRY The volume of the rectangular prism Teaching Tip Point out that
is given by V(x)  x3  3x2
36x  32. Find the the Factor Theorem does not
?
missing measures.
say anything about which
The volume of a rectangular prism is 艎  w  h. numbers to try. Techniques for
?
You know that one measure is x  4, so x  4 is a x
4 identifying potential factors will
factor of V(x). be introduced later in the
chapter.
4 1 3 36 32
4 28 32 3 GEOMETRY The volume of
1 7 8 0 the rectangular prism is given
The quotient is x2  7x  8. Use this to factor V(x). by V(x)  x3  7x2  2x  40.
V(x)  x3  3x2  36x  32 Volume function
Find the missing measures.
 (x  4)(x2  7x  8) Factor.

 (x  4)(x  8)(x  1) Factor the trinomial x2  7x – 8. ?

So, the missing measures of the prism are x  8 and x  1.

x
2
www.algebra2.com/extra_examples Lesson 7-4 The Remainder and Factor Theorems 367
?
The missing measures of the
rectangular prism are x  4 and
x  5.
Differentiated Instruction
Intrapersonal Have students describe two or three things about this
lesson that they found difficult to understand. Then have them address
each item by writing an explanation that will help them review the
material later or that they can refer to if they become confused in later
lessons. Suggest that students add these notes to their Study Notebooks.

Lesson 7-4 The Remainder and Factor Theorems 367

 3 Practice/Apply Concept Check 1. OPEN ENDED Give an example of a polynomial function that has a remainder
of 5 when divided by x  4. Sample answer: f (x)  x2
2x
3
2. State the degree of the depressed polynomial that is the result of dividing
3. dividend: x3  6x  x5  3x4  16x  48 by one of its first-degree binomial factors. 4
Study Notebook 32; divisor: x  2;
3. Write the dividend, divisor, quotient, 2 1 0 6 32
Have students— quotient: x2
2x  10; and remainder represented by the 2 4 20
• add the definitions/examples of remainder: 12 synthetic division at the right.
1 2 10 12
the vocabulary terms to their
Vocabulary Builder worksheets for Guided Practice Use synthetic substitution to find f(3) and f(
4) for each function.
GUIDED PRACTICE KEY 4. f(x)  x3  2x2  x  1 7,
91 5. f(x)  5x4  6x2  2 353, 1186
Chapter 7.
Exercises Examples
• write a note explaining how to Given a polynomial and one of its factors, find the remaining factors of the
4, 5 1 polynomial. Some factors may not be binomials.
determine whether a given 6
9 2
6. x3  x2  5x  3; x  1 x  1, x
3 7. x3  3x  2; x  1 x
1, x  2
binomial is a factor of a given 10 3
8. 6x3  25x2  2x  8; 3x  2 9. x4  2x3  8x  16; x  2
polynomial. 2x  1, x
4 x
2, x2  2x  4
• include any other item(s) that they Application For Exercises 10–12,
find helpful in mastering the skills use the graph at the right. USA TODAY Snapshots®
The projected sales of e-books
in this lesson. can be modeled by the function Digital book sales expected to grow
S(x)  17x3  200x2  113x  44, In the $20 billion publishing industry, e-books $2.4
account for less than 1% of sales now. But they billion
where x is the number of years are expected to claim 10% by 2005. Projected
since 2000. 10. $2.894 billion e-book sales:

10. Use synthetic substitution to $1.7

billion
About the Exercises… estimate the sales for 2006.

Organization by Objective 11. Evaluate S(6). $2.894 billion $1.1

billion
• Synthetic Substitution: 12. Which method—synthetic
13–20 division or direct substitution—
do you prefer to use to evaluate $445
• Factors of Polynomials: polynomials? Explain your
million
$131
21–30 answer. Sample answer: $41
million
million
Direct substitution, because
Odd/Even Assignments it can be done quickly with a
2000 2001 2002
Source: Andersen Consulting
2003 2004 2005

Exercises 13–30 are structured calculator. USA TODAY

so that students practice the

same concepts whether they ★ indicates increased difficulty
are assigned odd or even Practice and Apply
problems.
Homework Help Use synthetic substitution to find g(3) and g(
4) for each function. 18. 267, 680
Assignment Guide For
Exercises
See
Examples 13. g(x)  x2  8x  6
9, 54 14. g(x)  x3  2x2  3x  1 37,
19
Basic: 13–31 odd, 37, 38, 41, 13–20 1 15. g(x)  x3  5x  2 14,
42 16. g(x)  x4  6x  8 55, 272
21–36 2
46–62 37–44 3 17. g(x)  2x3  8x2  2x  5
19,
243 18. g(x)  3x4  x3  2x2  x  12
Average: 13–35 odd, 37–41, 19. g(x)  x5  8x3  2x  15 450,
1559 20. g(x)  x6  4x4  3x2  10 422, 3110
Extra Practice
46–62 See page 843.
Given a polynomial and one of its factors, find the remaining factors of the
Advanced: 14–36 even, 39–59 polynomial. Some factors may not be binomials. 24. x
3, x
1
(optional: 60–62) 21. x3  2x2  x  2; x  1 x  1, x  2 22. x3  x2  10x  8; x  1 x
4, x  2
23. x3  x2  16x  16; x  4 x
4, x  1 24. x3  6x2  11x  6; x  2
368 Chapter 7 Polynomial Functions
Answers
1
25. x  3, x
 or 2x
1
2 Online Lesson Plans
4
26. x
1, x   or 3x  4
3 USA TODAY Education’s Online site offers resources and
27. x  7, x
4 interactive features connected to each day’s newspaper.
28. x
1, x  6 Experience TODAY, USA TODAY’s daily lesson plan, is
available on the site and delivered daily to subscribers.
29. x
1, x 2  2x  3 This plan provides instruction for integrating USA TODAY
30. 2x
3, 2x  3, 4x 2  9 graphics and key editorial features into your mathematics
classroom. Log on to www.education.usatoday.com.

368 Chapter 7 Polynomial Functions

25–30. See margin. 25. 2x3  5x2  28x  15; x  5 26. 3x3  10x2  x  12; x  3
NAME ______________________________________________ DATE ____________ PERIOD _____

Study
7-4 Guide
Study andIntervention
Guide and Intervention,
27. 2x3  7x2  53x  28; 2x  1 28. 2x3  17x2  23x  42; 2x  7 p. 393 (shown)
The Remainder andTheorems
and Factor p. 394
29. x4  2x3  2x2  2x  3; x  1 30. 16x5  32x4  81x  162; x  2 Synthetic Substitution
Remainder The remainder, when you divide the polynomial f(x ) by (x  a), is the constant f(a).
Theorem f(x)  q(x) (x  a)  f(a), where q(x) is a polynomial with degree one less than the degree of f(x).

Example 1 If f(x)  3x4  2x3
5x2  x
2, find f(
2).

31. Use the graph of the polynomial f (x ) Method 1 Synthetic Substitution Method 2 Direct Substitution
By the Remainder Theorem, f(2) should Replace x with 2.
Changes in world function at the right to determine be the remainder when you divide the f(x)  3x4  2x3  5x2  x  2
polynomial by x  2.
f(2)  3(2)4  2(2)3  5(2)2  (2)  2
population can be at least one binomial factor of the 2 3 2 5 1 2  48  16  20  2  2 or 8
O x 6 8 6 10
modeled by a polynomial. Then find all the factors 3 4 3 5 8
So f(2)  8.

of the polynomial. x
2, x  2, x2  1
The remainder is 8, so f(2)  8.
polynomial function. Example 2 If f(x)  5x3  2x
1, find f(3).
Visit www.algebra2. 32. Use synthetic substitution to show that Again, by the Remainder Theorem, f(3) should be the remainder when you divide the
polynomial by x  3.
com/webquest to x  8 is a factor of x3  4x2  29x  24. f (x )  x 4
3 x 2
4 3 50 2 1
15 45 141
continue work on your Then find any remaining factors. 5 15 47 140
The remainder is 140, so f(3)  140.
WebQuest project.
See pp. 407A–407H; (x  3)(x  1). Exercises

Find values of k so that each remainder is 3. Use synthetic substitution to find f(
5) and f  for each function. 12 
★ 33. (x2  x  k)  (x  1) 3 ★ 34. (x2  kx  17)  (x  2) 8
3
1. f(x)  3x2  5x  1
101;  2. f(x)  4x2  6x  7 63;
3
4
35 29
3. f(x)  x3  3x2  5 195;
 4. f(x)  x4  11x2  1 899; 
★ 35. (x2  5x  7)  (x  k) 1, 4 ★ 36. (x3  4x2  x  k)  (x  2)
3 8 16

Lesson 7-4
Use synthetic substitution to find f(4) and f(
3) for each function.

5. f(x)  2x3  x2  5x  3 6. f(x)  3x3  4x  2

127;
27 178;
67
ENGINEERING For Exercises 37 and 38, use the following information. 7. f(x)  5x3  4x2  2 8. f(x)  2x4  4x3  3x2  x  6
258;
169 302; 288
When a certain type of plastic is cut into sections, the length of each section
9. f(x)  5x4  3x3  4x2  2x  4 10. f(x)  3x4  2x3  x2  2x  5
determines its strength. The function f(x)  x4  14x3  69x2  140x  100 can 1404; 298 627; 277

describe the relative strength of a section of length x feet. Sections of plastic x feet 11. f(x)  2x4  4x3  x2  6x  3
219; 282
12. f(x)  4x4  4x3  3x2  2x  3
805; 462
long, where f(x)  0, are extremely weak. After testing the plastic, engineers Gl NAME
/M G ______________________________________________
Hill 393 DATE ____________
GlPERIOD
Al _____
b 2

discovered that sections 5 feet long were extremely weak. Skills

7-4 Practice,
Practice (Average)
p. 395 and
Practice, p. 396 (shown)
37. Show that x  5 is a factor of the polynomial function. See pp. 407A–407H. The Remainder and Factor Theorems
Use synthetic substitution to find f(
3) and f(4) for each function.
1. f(x)  x2  2x  3 6, 27 2. f(x)  x2  5x  10 34, 6
38. Are there other lengths of plastic that are extremely weak? Explain your
3. f(x)  x2  5x  4 20,
8 4. f(x)  x3  x2  2x  3
27, 43
reasoning. Yes, 2-ft lengths; the binomial x
2 is a factor of the polynomial 5. f(x)  x3  2x2  5
4, 101 6. f(x)  x3  6x2  2x
87,
24
since f (2)  0. 7. f(x)  x3  2x2  2x  8
31, 32 8. f(x)  x3  x2  4x  4
52, 60

9. f(x)  x3  3x2  2x  50
56, 70 10. f(x)  x4  x3  3x2  x  12 42, 280

ARCHITECTURE For Exercises 39 and 40, use the following information.
11. f(x)  x4  2x2  x  7 73, 227 12. f(x)  2x4  3x3  4x2  2x  1 286, 377
Elevators traveling from one floor to the next do not travel at a constant speed. 13. f(x)  2x4  x3  2x2  26 181, 454 14. f(x)  3x4  4x3  3x2  5x  3 390, 537
Suppose the speed of an elevator in feet per second is given by the function 15. f(x)  x5  7x3  4x  10 16. f(x)  x6  2x5  x4  x3  9x2  20

f(t)  0.5t4  4t3  12t2  16t, where t is the time in seconds.
430, 1446 74, 5828

Given a polynomial and one of its factors, find the remaining factors of the
39. Find the speed of the elevator at 1, 2, and 3 seconds. 7.5 ft/s, 8 ft/s, 7.5 ft/s polynomial. Some factors may not be binomials.
17. x3  3x2  6x  8; x  2 18. x3  7x2  7x  15; x  1
x  1, x  4 x  3, x  5
40. It takes 4 seconds for the elevator to go from one floor to the next. Use synthetic
19. x3  9x2  27x  27; x  3 20. x3  x2  8x  12; x  3
substitution to find f(4). Explain what this means. 0; The elevator is stopped. x
3, x
3 x
2, x
2

21. x3  5x2  2x  24; x  2 22. x3  x2  14x  24; x  4

x  3, x  4 x
3, x
2

41. CRITICAL THINKING Consider the polynomial f(x)  ax4  bx3  cx2  dx  e, 23. 3x3  4x2  17x  6; x  2 24. 4x3  12x2  x  3; x  3
x
3, 3x
1 2x
1, 2x  1
where a  b  c  d  e  0. Show that this polynomial is divisible by x  1. 25. 18x3  9x2  2x  1; 2x  1 26. 6x3  5x2  3x  2; 3x  2
See margin. 3x  1, 3x
1 2x  1, x  1
27. x5  x4  5x3  5x2  4x  4; x  1 28. x5  2x4  4x3  8x2  5x  10; x  2
PERSONAL FINANCE For Exercises 42–45, use the following information. x
1, x  1, x
2, x  2 x
1, x  1, x2  5

Zach has purchased some home theater equipment for $2000, which he is financing 29. POPULATION The projected population in thousands for a city over the next several
years can be estimated by the function P(x)  x3  2x2  8x  520, where x is the

Architecture through the store. He plans to pay $340 per month and wants to have the balance number of years since 2000. Use synthetic substitution to estimate the population
for 2005. 655,000
paid off after six months. The formula B(x)  2000x6  340(x5  x4  x3  x2  x  1) 30. VOLUME The volume of water in a rectangular swimming pool can be modeled by the
The Sears Tower elevators polynomial 2x3  9x2  7x  6. If the depth of the pool is given by the polynomial

operate as fast as 1600 feet represents his balance after six months if x represents 1 plus the monthly interest 2x  1, what polynomials express the length and width of the pool? x
3 and x
2

per minute—among the rate (expressed as a decimal). Gl NAME

/M G ______________________________________________
Hill 396 DATE ____________
Gl PERIOD
Al _____
b 2
Reading
7-4 Readingto
to Learn
Learn Mathematics
fastest in the world. 42. Find his balance after 6 months if the annual interest rate is 12%. (Hint: The Mathematics, p. 397 ELL
The Remainder and Factor Theorems
Source: www.the-skydeck.com monthly interest rate is the annual rate divided by 12, so x  1.01.) $31.36 Pre-Activity How can you use the Remainder Theorem to evaluate polynomials?
Read the introduction to Lesson 7-4 at the top of page 365 in your textbook.
43. Find his balance after 6 months if the annual interest rate is 9.6%. $16.70 Show how you would use the model in the introduction to estimate the
number of international travelers (in millions) to the United States in the
year 2000. (Show how you would substitute numbers, but do not actually
44. How would the formula change if Zach wanted to pay the balance in five calculate the result.)
Sample answer: 0.02(14)3
0.6(14)2  6(14)  25.9
months? B(x)  2000x5
340(x4  x3  x2  x  1)
Reading the Lesson
45. Suppose he finances his purchase at 10.8% and plans to pay $410 every month. 1. Consider the following synthetic division.
Will his balance be paid in full after five months? No, he will still owe $4.40. 1 3 2
3
6
5 1
4

3 5 1 3
www.algebra2.com/self_check_quiz Lesson 7-4 The Remainder and Factor Theorems 369 a. Using the division symbol , write the division problem that is represented by this
synthetic division. (Do not include the answer.) (3x3  2x2
6x  4)  (x
1)

b. Identify each of the following for this division.

dividend 3x3  2x2
6x  4 divisor x
1

NAME ______________________________________________ DATE ____________ PERIOD _____ quotient 3x3  5x
1 remainder 3
Answer Enrichment,
7-4 Enrichment p. 398 c. If f(x)  3x3  2x2  6x  4, what is f(1)? 3

2. Consider the following synthetic division.

41. By the Remainder Theorem, the Using Maximum Values

3 1
3
0 0
9 27
27

remainder when f(x) is divided by x
1 is

1 3 9 0
Many times maximum solutions are needed for different situations. For
instance, what is the area of the largest rectangular field that can be enclosed a. This division shows that x3 is a factor of x3  27 .

equivalent to f(1), or a  b  c  d  e.
with 2000 feet of fencing?
Lesson 7-4

b. The division shows that
3 is a zero of the polynomial function

Let x and y denote the length and width f(x)  x3  27 .

Since a  b  c  d  e  0, the of the field, respectively.

Perimeter: 2x  2y  2000 → y  1000  x

y c. The division shows that the point
function f(x)  x3  27 .
(
3, 0) is on the graph of the polynomial

remainder when f(x) is divided by x
1 Area: A  xy  x(1000  x)  x2  1000x x

Helping You Remember
is 0. Therefore, x
1 is a factor of f(x). This problem is equivalent to finding
the highest point on the graph of
A
3. Think of a mnemonic for remembering the sentence, “Dividend equals quotient times
divisor plus remainder.”
A(x)  x2  1000x shown on the right.
Sample answer: Definitely every quiet teacher deserves proper rewards.
Complete the square for x2  1000x.

A  (x2  1000x  5002)  5002

 (x  500)2  5002

Lesson 7-4 The Remainder and Factor Theorems 369

 46. WRITING IN MATH Answer the question that was posed at the beginning of
4 Assess the lesson. See margin.
How can you use the Remainder Theorem to evaluate polynomials?
Include the following items in your answer:
Open-Ended Assessment
• an explanation of when it is easier to use the Remainder Theorem to evaluate
Speaking Ask students to offer a polynomial rather than substitution, and
a verbal comparison of the use of • evaluate the expression for the number of international travelers to the U.S.
synthetic substitution and the use for x  20.
of direct substitution to determine
Standardized 47. Determine the zeros of the function f(x)  x2  7x  12 by factoring. D
factors of a polynomial. Test Practice A 7, 12 B 3, 4 C 5, 5 D 4, 3
48. SHORT RESPONSE Using the graph f (x )
Getting Ready for of the polynomial function at the right, 2

4
2 O
Lesson 7-5 find all the factors of the polynomial
x5  x4  3x3  3x2  4x  4.
4
2 4x

PREREQUISITE SKILL In Lesson x
2, x  2, x  1, x2  1

8
7-5, students will find the zeros
of polynomial functions by using
12
the Quadratic Formula. Use
Exercises 60–62 to determine
your students’ familiarity with
using the Quadratic Formula. Maintain Your Skills
Mixed Review Write each expression in quadratic form, if possible. (Lesson 7-3)
Answers 49. x4  8x2  4 50. 9d6  5d3  2 51. r4  5r3  18r
(x 2)2
8(x 2)  4 9(d 3)2  5(d 3)
2 not possible
46. Using the Remainder Theorem you Graph each polynomial function. Estimate the x-coordinates at which the relative
maxima and relative minima occur. (Lesson 7-2) 52–53. See margin for graphs.
can evaluate a polynomial for a
value of a by dividing the polyno- 52. Sample answer: 52. f(x)  x3  6x2  4x  3 53. f(x)  x4  2x3  3x2  7x  4
rel. max. at x  0.5,
mial by x
a using synthetic rel. min. at x  3.5 54. PHYSICS A model airplane is fixed on a string so that it flies around in a circle.
division. Answers should include
 4 r 
2
53. Sample answer: The formula Fc  m

describes the force required to keep the airplane
the following. T2
rel. max. at x 
1 going in a circle, where m represents the mass of the plane, r represents the
• It is easier to use the Remainder and x  1.5, rel. min. radius of the circle, and T represents the time for a revolution. Solve this
Theorem when you have polyno- at x  1 formula for T. Write in simplest radical form. (Lesson 5-8) 2
 mrFc
T 

Fc
mials of degree 2 and lower or Solve each matrix equation. (Lesson 4-1)
when you have access to a 5a  2b 17
calculator.
55.   
7x
12

28
6y 
(4,
2) 56.
a  7b 
4  
(
3,
1) 
• The estimated number of Identify each function as S for step, C for constant, A for absolute value, or P for
international traveler to the piecewise. (Lesson 2-6)
U.S. in 2006 is 65.9 million. 57. y
A 58. y C 59. y S
52. f (x )
16
O x
f (x)  x 3
6x 2  4x  3
8

7 
17
60.

 O x
2 O x
9 
 57

2 O 2 4 x 61.


6

8
Getting Ready for PREREQUISITE SKILL Find the exact solutions of each equation by using the

16 the Next Lesson Quadratic Formula. (For review of the Quadratic Formula, see Lesson 6-5.)
3  i
7



60. x2  7x  8  0 61. 3x2  9x  2  0 62. 2x2  3x  2  0 4
53. f (x ) 370 Chapter 7 Polynomial Functions

4
2 O 2 4x

4

f (x) 
x 4  2x 3  3x 2
7x  4

370 Chapter 7 Polynomial Functions

 Roots and Zeros Lesson
Notes

• Determine the number and type of roots for a polynomial equation.

• Find the zeros of a polynomial function.
1 Focus
can the roots of an equation be used in pharmacology?
When doctors prescribe medication, they give patients
5-Minute Check
instructions as to how much to take and how often it should be Transparency 7-5 Use as a
taken. The amount of medication in your body varies with quiz or review of Lesson 7-4.
time. Suppose the equation M(t)  0.5t4  3.5t3  100t2  350t
models the number of milligrams of a certain medication in the Mathematical Background notes
bloodstream t hours after it has been taken. The doctor can use are available for this lesson on
the roots of this equation to determine how often the patient
should take the medication to maintain a certain concentration p. 344D.
in the body.
Building on Prior
TYPES OF ROOTS You have already learned that a zero of a function f(x) is Knowledge
any value c such that f(c)  0. When the function is graphed, the real zeros of the
function are the x-intercepts of the graph.
In Chapter 6, students learned
several methods for finding the
roots of quadratic equations. In
Zeros, Factors, and Roots this lesson, students will incorpo-
Let f(x)  an xn  …  a1x  a0 be a polynomial function. Then rate those techniques into finding
• c is a zero of the polynomial function f(x), roots of polynomial equations.
• x  c is a factor of the polynomial f(x), and
• c is a root or solution of the polynomial equation f(x)  0. can the roots of an
In addition, if c is a real number, then (c, 0) is an intercept of the graph of f(x).
equation be used in
pharmacology?
Study Tip Ask students:
When you solve a polynomial equation with degree greater than zero, it may have
one or more real roots, or no real roots (the roots are imaginary numbers). Since real • Would the given equation be
Look Back
For review of complex numbers and imaginary numbers both belong to the set of complex numbers, all valid for any value of t? Explain.
numbers, see Lesson 5-9. polynomial equations with degree greater than zero will have at least one root in the No; the value of t must be positive
set of complex numbers. This is the Fundamental Theorem of Algebra .
because the number of hours cannot
be negative. Also, once the number
Fundamental Theorem of Algebra of milligrams, M(t), reaches zero as
Every polynomial equation with degree greater than zero has at least one root in the value of t increases, any greater
the set of complex numbers.
values of t will be meaningless also.
• What would a root of this
Example 1 Determine Number and Type of Roots equation tell the doctor?
There is no more medication in the
Solve each equation. State the number and type of roots.
patient’s bloodstream.
a. x  3  0
x30 Original equation
x  3 Subtract 3 from each side.
This equation has exactly one real root, 3.
Lesson 7-5 Roots and Zeros 371

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 7 Resource Masters
• Study Guide and Intervention, pp. 399–400 5-Minute Check Transparency 7-5
• Skills Practice, p. 401 Answer Key Transparencies
• Practice, p. 402
• Reading to Learn Mathematics, p. 403 Technology
• Enrichment, p. 404 Alge2PASS: Tutorial Plus, Lesson 14
• Assessment, pp. 443, 445 Interactive Chalkboard

Lesson x-x Lesson Title 371

 b. x2
8x  16  0
2 Teach x2  8x  16  0
(x  4)2  0
Original equation
Factor the left side as a perfect square trinomial.
Study Tip
TYPES OF ROOTS Reading Math
x4 Solve for x using the Square Root Property.

In addition to double Since x  4 is twice a factor of x2  8x  16, 4 is a double root. So this equation
In-Class Example Power
Point® roots, equations can have has two real roots, 4 and 4.
triple or quadruple roots.
1 Solve each equation. State the In general, these roots are c. x3  2x  0
referred to as repeated
number and type of roots. roots. x3  2x  0 Original equation
x(x2  2)  0 Factor out the GCF.
a. a  10  0 This equation has
exactly one real root, 10. Use the Zero Product Property.

b. x2  2x  48  0 This equation x0 or x2  2  0

has two real roots, 6 and
8. x2  2 Subtract two from each side.
x  2 or  i2 Square Root Property
c. 3a3  18a  0 This equation has
one real root, 0, and two imagi-  and i2.
This equation has one real root, 0, and two imaginary roots, i2
nary roots i
6 and
i
6. d. x4
1  0
d. y4  16  0 This equation has x4  1  0
two real roots, 2 and
2, and (x2 1) (x2  1)  0
two imaginary roots, 2i and
2i. (x  1) (x  1)(x  1)  0
2

x2  1  0 or x10 or x10
x2  1 x  1 x1
x  1 or  i
This equation has two real roots, 1 and 1, and two imaginary roots, i and i.

Compare the degree of each equation and the number of roots of each equation in
Example 1. The following corollary of the Fundamental Theorem of Algebra is an
even more powerful tool for problem solving.

Corollary
Descartes A polynomial equation of the form P(x)  0 of degree n with complex coefficients
René Descartes has exactly n roots in the set of complex numbers.
(1596–1650) was a French
mathematician and
Similarly, a polynomial function of nth degree has exactly n zeros.
philosopher. One of his
best-known quotations
comes from his Discourse French mathematician René Descartes made more discoveries about zeros of
on Method: “I think, polynomial functions. His rule of signs is given below.
therefore I am.”
Source: A History of Mathematics
Descartes’ Rule of Signs
If P(x) is a polynomial with real coefficients whose terms are arranged in
descending powers of the variable,
• the number of positive real zeros of y  P(x) is the same as the number of
changes in sign of the coefficients of the terms, or is less than this by an even
number, and
• the number of negative real zeros of y  P(x) is the same as the number of
changes in sign of the coefficients of the terms of P(x), or is less than this
number by an even number.

372 Chapter 7 Polynomial Functions

Teacher to Teacher
Warren Zarrell James Monroe H.S., North Hills, CA
“My students have difficulty finding p(–x), as found in Example 2. I tell them
to change the sign of every odd degree term in the polynomial.”

372 Chapter 7 Polynomial Functions

 Example 2 Find Numbers of Positive and Negative Zeros In-Class Example Power
Point®
State the possible number of positive real zeros, negative real zeros, and
imaginary zeros of p(x)  x5
6x4
3x3  7x2
8x  1. Teaching Tip Point out to stu-
Since p(x) has degree 5, it has five zeros. However, some of them may be dents the method for determin-
imaginary. Use Descartes’ Rule of Signs to determine the number and type of real ing the number of imaginary
zeros. Count the number of changes in sign for the coefficients of p(x). zeros for a polynomial function.
p(x)  x5  6x4  3x3  7x2  8x  1 In Example 2, the polynomial has
degree 5, so it has a maximum
yes no yes yes yes of 5 real zeros. You find the num-
 to   to   to   to   to  bers of positive and negative
Since there are 4 sign changes, there are 4, 2, or 0 positive real zeros. real zeros, and subtract the sum
of these two numbers from 5 to
Find p(x) and count the number of changes in signs for its coefficients.
find the number of imaginary
p(x)  (x)5  6(x)4  3(x)3  7(x)2  8(x)  1 zeros. Remind students that
 x5  6x4  3x3  7x2  8x  1 imaginary zeros come in conju-
gate pairs, so the number of
no yes no no no imaginary zeros must be an
 to   to   to   to   to 
even number.
Study Tip Since there is 1 sign change, there is exactly 1 negative real zero.
Zero at the Origin Thus, the function p(x) has either 4, 2, or 0 positive real zeros and exactly 1 negative
2 State the possible number of
Recall that the number 0 real zero. Make a chart of the possible combinations of real and imaginary zeros. positive real zeros, negative
has no sign. Therefore, if real zeros, and imaginary
0 is a zero of a function,
the sum of the number
Number of Positive Number of Negative Number of Total Number zeros of p(x)  x6  4x3 
of positive real zeros,
Real Zeros Real Zeros Imaginary Zeros of Zeros
2x2  x  1. The function has
4 1 0 4105
negative real zeros, and either 2 or 0 positive real zeros,
imaginary zeros is 2 1 2 2125
reduced by how many 2 or 0 negative real zeros, and 6,
0 1 4 0145
times 0 is a zero of the 4, or 2 imaginary zeros.
function.

FIND ZEROS We can find all of the zeros of a function using some of the FIND ZEROS
strategies you have already learned.
In-Class Example Power
Point®
Example 3 Use Synthetic Substitution to Find Zeros
Find all of the zeros of f(x)  x3
4x2  6x
4. 3 Find all of the zeros of
Since f(x) has degree 3, the function has three zeros. To determine the possible f(x)  x3  x2  2x  4. The
number and type of real zeros, examine the number of sign changes for f(x) function has one real zero at
and f(x). x 
1, and two imaginary
f(x)  x3  4x2  6x  4 f(x)  x3  4x2  6x  4 zeros at x  1  i
3 and
x  1
i
3.
yes yes yes no no no

Since there are 3 sign changes for the coefficients of f(x), the function has 3 or 1
positive real zeros. Since there are no sign changes for the coefficient of f(x), f(x)
has no negative real zeros. Thus, f(x) has either 3 real zeros, or 1 real zero and 2
imaginary zeros.
To find these zeros, first list some possibilities and then eliminate those that are
not zeros. Since none of the zeros are negative and evaluating the function for 0
results in –4, begin by evaluating f(x) for positive integral values from 1 to 4.
You can use a shortened form of synthetic substitution to find f(a) for several
values of a.
www.algebra2.com/extra_examples Lesson 7-5 Roots and Zeros 373

Differentiated Instruction
Kinesthetic As you work Example 3 and Guided Practice Exercises 8–11
in class, provide each student with approximately 20 slips of paper. As stu-
dents begin the process of finding the zeros of each polynomial function,
have them first determine from the degree of the polynomial the
number of zeros they need to find. Students should then count off slips
of paper, one for each zero. As students work the problem and find the
zeros, they should record the information about each zero (positive,
negative, imaginary) on one of the slips of paper.
Lesson 7-5 Roots and Zeros 373
In-Class Example Power Study Tip x 1
4 6
4
Point®
Finding Zeros 1 1 3 3 1 Each row in the table shows the
coefficients of the depressed
4 Write a polynomial function While direct substitution 2 1 2 2 0
could be used to find each polynomial and the remainder.
of least degree with integral 3 1 1 3 5
real zero of a polynomial,
coefficients whose zeros using synthetic 4 1 0 6 20
substitution provides
include 4 and 4  i. you with a depressed From the table, we can see that one zero occurs at x  2. Since the depressed
f(x)  x 3
12x 2  49x
68 is polynomial that can polynomial of this zero, x2  2x  2, is quadratic, use the Quadratic Formula to
a polynomial function of least be used to find any
find the roots of the related quadratic equation, x2  2x  2  0.
imaginary zeros.
degree with integral coefficients b   
b2  4ac
whose zeros are 4, 4
i, and x 

Quadratic Formula
2a
4  i. (2)    
(2)2  
4(1)(2)




2(1)
Replace a with 1, b with 2, and c with 2.




2  4
Simplify.
2
2  2i


4  1  2i
2
1i Simplify.

Thus, the function has one real zero f (x )

at x  2 and two imaginary zeros at
x  1  i and x  1  i. The graph of
the function verifies that there is only x
O
one real zero.
f (x )  x 3
4 x 2  6 x
4

In Chapter 6, you learned that solutions of a quadratic equation that contains

imaginary numbers come in pairs. This applies to the zeros of polynomial functions
as well. For any polynomial function, if an imaginary number is a zero of that
function, its conjugate is also a zero. This is called the Complex Conjugates
Theorem .

Complex Conjugates Theorem

Suppose a and b are real numbers with b  0. If a  bi is a zero of a polynomial
function with real coefficients, then a  bi is also a zero of the function.

Standardized Example 4 Use Zeros to Write a Polynomial Function

Test Practice Short-Response Test Item

Write a polynomial function of least degree with integral coefficients whose

zeros include 3 and 2  i.

Read the Test Item

• If 2  i is a zero, then 2  i is also a zero according to the Complex Conjugates
Theorem. So, x  3, x  (2  i), and x  (2  i) are factors of the polynomial
function.
Solve the Test Item
• Write the polynomial function as a product of its factors.
f(x)  (x  3)[x  (2  i)][x  (2  i)]
374 Chapter 7 Polynomial Functions

Standardized Example 4 Ask students to indicate with a show of hands how

many of them have made mistakes in mathematics exercises
Test Practice because they could not read their own handwriting. Stress that
throughout this course, students must work using neat and
careful handwriting. It is extremely easy to misread coefficients and exponents, or misread i
as the number one. In addition, when answering short-response items on standardized
tests, students must be aware that if their handwriting is illegible or difficult to read then
their answers will not be graded or they may be penalized.

374 Chapter 7 Polynomial Functions

 • Multiply the factors to find the polynomial function.

f(x)  (x  3)[x  (2  i)][x  (2  i)] Write an equation. 3 Practice/Apply

 (x  3)[(x  2)  i][(x  2)  i] Regroup terms.

Test-Taking Tip  (x  3)[(x  2)2  i2] Rewrite as the difference of two squares.

Knowing quadratic  (x  3)[x2  4x  4  (1)] Square x  2 and replace i 2 with –1.

identities like the  (x  3)(x2  4x  5) Simplify.
Study Notebook
difference of two squares
and perfect square  x3  4x2  5x  3x2  12x  15 Multiply using the Distributive Property. Have students—
trinomials can save you  x3  7x2  17x  15 Combine like terms. • add the definitions/examples of
time on standardized tests.
f(x)  x3  7x2  17x  15 is a polynomial function of least degree with integral the vocabulary terms to their
coefficients whose zeros are 3, 2  i, and 2  i. Vocabulary Builder worksheets for
1. Sample answer: p(x)  x3
6x2  x  1; p(x) has either 2 or 0 positive real Chapter 7.
zeros, 1 negative real zero, and 2 or 0 imaginary zeros. • summarize what they know so far
about identifying types of zeros
and finding some of them.
Concept Check 1. OPEN ENDED Write a polynomial function p(x) whose coefficients have two
sign changes. Then describe the nature of its zeros. • include any other item(s) that they
2. Explain why an odd-degree function must always have at least one real root. find helpful in mastering the skills
See margin. in this lesson.
3. State the least degree a polynomial equation with real coefficients can have if it
has roots at x  5  i, x  3  2i, and a double root at x  0. 6

Guided Practice Solve each equation. State the number and type of roots. 5.
7, 0, and 3; 3 real
GUIDED PRACTICE KEY 4. x2  4  0
2i; 2 imaginary 5. x3  4x2  21x  0
Exercises Examples
State the possible number of positive real zeros, negative real zeros, and About the Exercises…
4, 5 1 imaginary zeros of each function.
6, 7 2 Organization by Objective
8–11 3 6. f(x)  5x3  8x2  4x  3 7. r(x)  x5  x3  x  1 • Types of Roots: 13–24
12 4 2 or 0; 1; 2 or 0 2 or 0; 1; 2 or 4
• Find Zeros: 25–40
Find all of the zeros of each function.
8.
4, 1  2i, 1
2i 8. p(x)  x3  2x2  3x  20 9. f(x)  x3  4x2  6x  4 Odd/Even Assignments
9. 2, 1  i, 1
i 10. v(x)  x3  3x2  4x  12 2i,
2i, 3 11. f(x)  x3  3x2  9x  13 Exercises 13–40 are structured
2  3i, 2
3i,
1 so that students practice the
Standardized 12. SHORT RESPONSE Write a polynomial function of least degree with integral same concepts whether they
Test Practice coefficients whose zeros include 2 and 4i. f(x)  x3
2x2  16x
32 are assigned odd or even
problems.
★ indicates increased difficulty 15. 0, 3i,
3i; 1 real, 2 imaginary 16. 3i, 3i,
3i, and
3i; 4 imaginary
Practice and Apply Assignment Guide
Basic: 13–41 odd, 49–70
Homework Help Solve each equation. State the number and type of roots. 
5  i
71
8

; Average: 13–41 odd, 42–45,
For See 13. 3x  8  0
; 1 real 14. 2x2  5x  12  0 4
Exercises Examples 3 2 imaginary
13–18 1 15. x  9x  0
3 16. x  81  0
4 49–70
19–24, 41 2
17. x4  16  0 18. x5  8x3  16x  0 Advanced: 14–40 even, 44–66
25–34, 3
44–48 2,
2, 2i, and
2i; 2 real, 2 imaginary
2,
2, 0, 2, and 2, 5 real (optional: 67–70)
35–40, 42, 4 State the possible number of positive real zeros, negative real zeros, and
43 imaginary zeros of each function. 19–24. See margin.
Extra Practice 19. f(x)  x3  6x2  1 20. g(x)  5x3  8x2  4x  3
Answers
See page 843. 21. h(x)  4x3  6x2  8x  5 22. q(x)  x4  5x3  2x2  7x  9
23. p(x)  x5  6x4  3x3  7x2  8x  1 24. f(x)  x10  x8  x6  x4  x2  1 2. An odd-degree function approaches
www.algebra2.com/self_check_quiz Lesson 7-5 Roots and Zeros 375
positive infinity in one direction
and negative infinity in the other
direction, so the graph must cross
the x-axis at least once, giving it
Unlocking Misconceptions at least one real root.
Finding Zeros Students may incorrectly assume that they now know 19. 2 or 0; 1; 2 or 0
how to find all zeros. However, in this lesson they are using the guess- 20. 2 or 0; 1; 2 or 0
and-check technique to test possible zeros. In later lessons, students will 21. 3 or 1; 0; 2 or 0
learn further techniques for finding the zeros of polynomial functions.
22. 1; 3 or 1; 2 or 0
23. 4, 2, or 0; 1; 4, 2, or 0
24. 5, 3, or 1; 5, 3, or 1; 0, 2, 4, 6, or 8

Lesson 7-5 Roots and Zeros 375

 Study
NAME ______________________________________________ DATE

Guide andIntervention
Intervention,
____________ PERIOD _____
25.
2,
2  Find all of the zeros of each function.
7-5 Study Guide and
p. 399 (shown)
Roots and Zeros and p. 400 3i,
2i
3i 25. g(x)  x3  6x2  21x  26 26. h(x)  x3  6x2  10x  8
Types of Roots
•
The following statements are equivalent for any polynomial function f(x).
c is a zero of the polynomial function f(x).
26. 4, 1  i, 1
i 27. h(x)  4x4  17x2 4 28. f(x)  x3  7x2  25x  175
(x  c) is a factor of the polynomial f(x). i i
27. 2i,
2i,

,



•
• c is a root or solution of the polynomial equation f(x)  0. 29. g(x)  2x3  x2  28x  51 30. q(x)  2x3  17x2  90x  41
If c is real, then (c, 0) is an intercept of the graph of f(x). 2 2
Fundamental
Theorem of Algebra
Every polynomial equation with degree greater than zero has at least one root in the set
of complex numbers. 28. 5i,
5i, 7 31. f(x)  x3  5x2  7x  51 32. p(x)  x4  9x3  24x2  6x  40
3 33. r(x)  x4  6x3  12x2  6x  13 34. h(x)  x4  15x3  70x2  70x  156
Corollary to the A polynomial equation of the form P (x)  0 of degree n with complex coefficients has
Fundamental exactly n roots in the set of complex numbers.
29.
, 1  4i, 1
4i
3
2i, 3  2i,
1, 1 5
i, 5  i,
1, 6
Theorem of Algebras
If P (x) is a polynomial with real coefficients whose terms are arranged in descending 2
1
30. , 4  5i, 4
5i
powers of the variable,
• the number of positive real zeros of y  P(x) is the same as the number of changes in
Descartes’ Rule
of Signs
sign of the coefficients of the terms, or is less than this by an even number, and
2 Write a polynomial function of least degree with integral coefficients that has the
• the number of negative real zeros of y  P(x) is the same as the number of changes in
given zeros. 35–40. See margin.
31. 4
i, 4  i,
3
sign of the coefficients of the terms of P(x), or is less than this number by an even
number.

35. 4, 1, 5 36. 2, 2, 4, 6 37. 4i, 3, 3

Example 1 Solve the equation
6x3  3x  0 and state the
Example 2 State the number of positive
real zeros, negative real zeros, and imaginary
32. 3
i, 3  i, 4,
1
number and type of roots.
6x3  3x  0
zeros for p(x)  4x4
3x3  x2  2x
5.
Since p(x) has degree 4, it has 4 zeros.
38. 2i, 3i, 1 39. 9, 1  2i 40. 6, 2  2i
3x(2x2  1)  0 Use Descartes’ Rule of Signs to determine the
Use the Zero Product Property. number and type of real zeros. Since there are three
sign changes, there are 3 or 1 positive real zeros.
3x  0 or 2x2  1  0
x  0 or 2x2  1
Find p(x) and count the number of changes in 41. Sketch the graph of a polynomial function that has the indicated number and
sign for its coefficients.
x  

i2
2

p(x)  4(x)4  3(x)3  (x)2  2(x)  5 type of zeros. See pp. 407A–407H.
 4x4  3x3  x2  2x  5
The equation has one real root, 0,
and two imaginary roots, 
.
i2
2

Since there is one sign change, there is exactly 1
negative real zero. a. 3 real, 2 imaginary b. 4 real c. 2 imaginary
Exercises
Solve each equation and state the number and type of roots. SCULPTING For Exercises 42 and 43, use the following information.
1. x2  4x  21 0 2. 2x3  50x  0 3. 12x3  100x  0
3,
7; 2 real 0, 5; 3 real 5i
3
0,   
; 1 real, 2 imaginary
Antonio is preparing to make an ice sculpture. He 3 ft
3
State the number of positive real zeros, negative real zeros, and imaginary zeros has a block of ice that he wants to reduce in size by
Lesson 7-5

for each function.

4. f(x)  3x3  x2  8x  12 1; 2 or 0; 0 or 2
shaving off the same amount from the length, width,
5. f(x)  2x4  x3  3x  7 2 or 0; 0; 2 or 4 and height. He wants to reduce the volume of the ice 4 ft
6. f(x)  3x5  x4  x3  6x2  5 3 or 1; 2 or 0; 0, 2, or 4
block to 24 cubic feet. 42. (3
x)(4
x)(5
x)  24
Gl NAME
/M G ______________________________________________
Hill 399 DATE ____________
GlPERIOD
Al _____
b 2

Skills
7-5 Practice,
Practice p. 401 and 42. Write a polynomial equation to model this situation.
Practice,
(Average)
Roots and p. 402 (shown) Space Exploration 5 ft
Zeros
A space shuttle is a
43. How much should he take from each dimension? 1 ft
Solve each equation. State the number and type of roots.
1. 9x  15  0
5
2. x4  5x2 40 reusable vehicle, launched

; 1 real
1, 1,
2, 2; 4 real
3 like a rocket, which can put SPACE EXPLORATION For Exercises 44 and 45, use the following information.
3. x5  81x 4. x3  x2  3x  3  0

0,
3, 3,
3i, 3i; 3 real, 2 imaginary
1,

3

,
3
; 3 real
people and equipment in The space shuttle has an external tank for the fuel that the main
5. x3  6x  20  0 6. x4  x3  x2  x  2  0
orbit around Earth. The engines need for the launch. This tank is shaped like a capsule,

2, 1  3i; 1 real, 2 imaginary 2,
1,
i, i; 2 real, 2 imaginary
first space shuttle was a cylinder with a hemispherical dome at either end. The r
State the possible number of positive real zeros, negative real zeros, and launched in 1981. cylindrical part of the tank has an approximate volume of
imaginary zeros of each function.
Source: kidsastronomy.about.com 336 cubic meters and a height of 17 meters more than the
7. f(x)  4x3  2x2  x  3 8. p(x)  2x4  2x3  2x2  x  1
2 or 0; 1; 2 or 0 3 or 1; 1; 2 or 0
radius of the tank. (Hint: V(r)  r2h)
9. q(x)  3x4  x3  3x2  7x  5 10. h(x)  7x4  3x3  2x2  x  1 h
2 or 0; 2 or 0; 4, 2, or 0 2 or 0; 2 or 0; 4, 2, or 0 44. Write an equation that represents the volume of the
Find all the zeros of each function. cylinder. V(r)  r 3  17r 2
11. h(x)  2x3  3x2  65x  84 12. p(x)  x3  3x2  9x  7

7, , 4
3
2
1, 1  i
6
, 1
i
6
 45. What are the dimensions of the tank?
13. h(x)  x3  7x2  17x  15 14. q(x)  x4  50x2  49 radius  4 m, height  21 m
3, 2  i, 2
i
i, i,
7i, 7i
15. g(x)  x4  4x3  3x2  14x  8 16. f(x)  x4  6x3  6x2  24x  40

1,
1, 2,
4
2, 2, 3
i, 3  i

MEDICINE For Exercises 46–48, use the following information.
Write a polynomial function of least degree with integral coefficients that has the
given zeros. Doctors can measure cardiac output in patients at high risk for a heart attack by
17. 5, 3i 18. 2, 3  i monitoring the concentration of dye injected into a vein near the heart. A normal
f(x)  x 3  5x 2  9x  45 f(x)  x 3
4x 2
2x  20
19. 1, 4, 3i 20. 2, 5, 1  i
heart’s dye concentration is given by d(x)  0.006x4  0.15x3  0.05x2  1.8x,
f(x)  x 4
3x 3  5x 2
27x
36 f(x)  x 4
9x 3  26x 2
34x  20 where x is the time in seconds.
★ 46. How many positive real zeros, negative real zeros, and imaginary zeros exist for
21. CRAFTS Stephan has a set of plans to build a wooden box. He wants to reduce the
volume of the box to 105 cubic inches. He would like to reduce the length of each
dimension in the plan by the same amount. The plans call for the box to be 10 inches by
8 inches by 6 inches. Write and solve a polynomial equation to find out how much
Stephen should take from each dimension. (10
x)(8
x)(6
x)  105; 3 in.
this function? (Hint: Notice that 0, which is neither positive nor negative, is a
Gl NAME
/M G ______________________________________________
Hill 402 DATE ____________
Gl PERIOD
Al _____
b 2
zero of this function since d(0)  0.) 1; 2 or 0; 2 or 0
Reading
7-5 Readingto
to Learn
Learn Mathematics ★ 47. Approximate all real zeros to the nearest tenth by graphing the function using a
Mathematics, p. 403 ELL
Roots and Zeros graphing calculator. See margin for graph;
24.1,
4.0, 0, and 3.1.
Pre-Activity How can the roots of an equation be used in pharmacology? 49. Sample answer:
Read the introduction to Lesson 7-5 at the top of page 371 in your textbook.
f(x)  x
6x 
3 2 ★ 48. What is the meaning of the roots in this problem? Nonnegative roots represent
Using the model given in the introduction, write a polynomial equation
with 0 on one side that can be solved to find the time or times at which
there is 100 milligrams of medication in a patient’s bloodstream. 5x  12 and g(x)  times when there is no concentration of dye registering on the monitor.
0.5t 4  3.5t 3
100t 2  350t
100  0
2x3
12x2  10x  49. CRITICAL THINKING Find a counterexample to disprove the following
Reading the Lesson 24; each have zeros at statement.
1. Indicate whether each statement is true or false.
a. Every polynomial equation of degree greater than one has at least one root in the set
x  4, x 
2, and The polynomial function of least degree with integral coefficients with zeros at x  4,
of real numbers. false
b. If c is a root of the polynomial equation f(x)  0, then (x  c) is a factor of the
x  3. x  1, and x  3, is unique.
polynomial f(x). true
c. If (x  c) is a factor of the polynomial f(x), then c is a zero of the polynomial
function f. false
376 Chapter 7 Polynomial Functions
d. A polynomial function f of degree n has exactly (n  1) complex zeros. false

2. Let f(x)  x6  2x5  3x4  4x3  5x2  6x  7.

a. What are the possible numbers of positive real zeros of f ? 5, 3, or 1

NAME ______________________________________________ DATE ____________ PERIOD _____
b. Write f(x) in simplified form (with no parentheses).
x 6  2x 5  3x 4  4x 3  5x 2
6x
7
Enrichment,
7-5 Enrichment p. 404
Answers
What are the possible numbers of negative real zeros of f ? 1
c. Complete the following chart to show the possible combinations of positive real zeros,
negative real zeros, and imaginary zeros of the polynomial function f. The Bisection Method for Approximating Real Zeros 35. f(x)  x 3
2x 2
19x  20
Number of Number of Number of Total Number

36. f(x)  x 4
10x 3  20x 2  40x
96

The bisection method can be used to approximate zeros of polynomial
Positive Real Zeros Negative Real Zeros Imaginary Zeros of Zeros
functions like f (x)  x3  x2  3x  3.
5 1 0 6
Since f (1)  4 and f (2)  3, there is at least one real zero between 1 and 2.
3
1
1
1
2
4
6
6
12
The midpoint of this interval is

 1.5. Since f(1.5)  1.875, the zero is
2
1.5  2
between 1.5 and 2. The midpoint of this interval is

 1.75. Since
37. f(x)  x 4  7x 2
144
2

Helping You Remember

f(1.75) is about 0.172, the zero is between 1.5 and 1.75. The midpoint of this
1.5  1.75
interval is

 1.625 and f(1.625) is about 0.94. The zero is between
38. f(x)  x 5
x 4  13x 3
13x 2  36x
36
2
3. It is easier to remember mathematical concepts and results if you relate them to each 1.625  1.75

39. f(x)  x 3
11x 2  23x
45

other. How can the Complex Conjugates Theorem help you remember the part of 1.625 and 1.75. The midpoint of this interval is

 1.6875. Since
2
Descartes’ Rule of Signs that says, “or is less than this number by an even number.” f (1.6875) is about 0.41, the zero is between 1.6875 and 1.75. Therefore, the
Sample answer: For a polynomial function in which the polynomial has zero is 1.7 to the nearest tenth.

40. f(x)  x 3
10x 2  32x
48

real coefficients, imaginary zeros come in conjugate pairs. Therefore, there
must be an even number of imaginary zeros. For each pair of imaginary The diagram below summarizes the results obtained by the bisection method.
zeros, the number of positive or negative zeros decreases by 2.
sign of f (x ): – – – – + +

376 Chapter 7 Polynomial Functions

 50. CRITICAL THINKING If a sixth-degree polynomial equation has exactly five
distinct real roots, what can be said of one of its roots? Draw a graph of this
situation. One root is a double root; see margin for sample graph. 4 Assess
51. WRITING IN MATH Answer the question that was posed at the beginning of Open-Ended Assessment
the lesson. See pp. 407A–407H.
How can the roots of an equation be used in pharmacology? Writing Have students summa-
Include the following items in your answer:
rize the method for determining
• an explanation of what the roots of this equation represent, and
the number of real zeros for a
• an explanation of what the roots of this equation reveal about how often a
polynomial function, and how to
patient should take this medication. determine how many of them are
positive, negative, and imaginary.
Standardized 52. The equation x4  1  0 has exactly ____
?___ complex root(s). A
Test Practice A 4 B 0 C 2 D 1
Getting Ready for
53. How many negative real zeros does f(x)  x5  2x4  4x3  4x2  5x  6 have? C Lesson 7-6
A 3 B 2 C 1 D 0 BASIC SKILL In Lesson 7-6, stu-
dents will be introduced to the
Maintain Your Skills Rational Zero Theorem. They will
factor integral coefficients in order
Mixed Review Use synthetic substitution to find f(3) and f(4) for each function. (Lesson 7-4) to list every possible rational zero
54. f(x)  x3  5x2  16x  7 55. f(x)  x4  11x3  3x2  2x  5 of a polynomial. Use Exercises

127, 41
254, 915 67–70 to determine your students’
56. RETAIL The store Bunches of Boxes and Bags assembles boxes for mailing. The
store manager found that the volume of a box made from a rectangular piece
familiarity with finding all
of cardboard with a square of length x inches cut from each corner is possible rational values of an
4x3  168x2  1728x cubic inches. If the piece of cardboard is 48 inches long, a
expression 
, given all the
what is the width? (Lesson 7-3) 36 in. b
possible values of a and b.
Determine whether each function has a maximum or a minimum value. Then find
the maximum or minimum value of each function. (Lesson 6-1)
57. f(x)  x2  8x  3 58. f(x)  3x2  18x  5 59. f(x)  7  4x2 Assessment Options
min.;
13 max.; 32 min.;
7 Quiz (Lessons 7-4 and 7-5) is
Factor completely. If the polynomial is not factorable, write prime. (Lesson 5-4)
available on p. 443 of the Chapter 7
60. 15a2b2  5ab2c2 61. 12p2  64p  45 62. 4y3  24y2  36y Resource Masters.
5ab2(3a
c2) (6p
5)(2p
9) 4y(y  3)2
Use matrices A, B, C, and D to find the following. (Lesson 4-2) Mid-Chapter Test (Lessons 7-1
through 7-5) is available on
       

4 4 7 0
4
5 1
2
A 2
3 B 4 C
3 D
1 p. 445 of the Chapter 7 Resource
 
1 1 1

3 2
1 5 6
2 2 3
3 4 Masters.
63. 3
4

2 9 63. A  D 64. B  C 65. 3B  2A

 
11 5 66. Write an inequality for the graph y
64. 7 0 at the right. (Lesson 2-7)
4
5 2
y

x
1
3

 
29
8 O x

65. 8 9
16
16

a
Getting Ready for BASIC SKILL Find all values of 

given each replacement set.
b
the Next Lesson 67. a  {1, 5}; b  {1, 2} 68. a  {1, 2}; b  {1, 2, 7, 14}
67–70. See margin. 69. a  {1, 3}; b  {1, 3, 9} 70. a  {1, 2, 4}; b  {1, 2, 4, 8, 16}
Lesson 7-5 Roots and Zeros 377

Answers
47. 50. Sample graph: 1 5
f (x ) 67.   , 1,   , 5
2 2
1 1 2 1
68.   ,   ,   ,   , 1, 2
14 7 7 2
1 1
O x 69.   ,   , 1, 3
9 3
1 1 1 1
[
30, 10] scl: 5 by [
20, 20] scl: 5 70.   ,   ,   ,   , 1, 2, 4
16 8 4 2

Lesson 7-5 Roots and Zeros 377

 Lesson Rational Zero Theorem
Notes

• Identify the possible rational zeros of a polynomial function.

1 Focus • Find all the rational zeros of a polynomial function.

can the Rational Zero Theorem solve

5-Minute Check problems involving large numbers?
Transparency 7-6 Use as a
On an airplane, carry-on baggage must fit into the
quiz or review of Lesson 7-5. overhead compartment above the passenger’s seat.
The length of the compartment is 8 inches longer h
Mathematical Background notes than the height, and the width is 5 inches shorter
are available for this lesson on than the height. The volume of the compartment is
p. 344D. 2772 cubic inches. You can solve the polynomial
h8 h
5
equation h(h  8)(h  5)  2772, where h is the height,
h  8 is the length, and h  5 is the width, to find the
can the Rational Zero dimensions of the overhead compartment in which your luggage must fit.
Theorem solve prob-
lems involving large numbers?
Ask students: IDENTIFY RATIONAL ZEROS Usually it is not practical to test all possible
zeros of a polynomial function using only synthetic substitution. The Rational Zero
• Why does the polynomial Theorem can help you choose some possible zeros to test.
equation contain an expression
in which the variable h appears Rational Zero Theorem
three times? The actual width and • Words Let f(x)  a0xn  a1xn  1  …  an  2x2  an  1x  an represent a
length of the overhead compart- p
polynomial function with integral coefficients. If

is a rational
q
ment are not known; only their
number in simplest form and is a zero of y  f(x), then p is a factor of
relationships to the height of the an and q is a factor of a0.
compartment are known. • Example Let f(x)  2x3  3x2  17x  12. If
32
is a zero of f(x), then 3 is a factor
• How do you know that h  8 is of 12 and 2 is a factor of 2.
the length? The length is 8 inches
longer than the height, which is In addition, if the coefficient of the x term with the highest degree is 1, we have
represented by the variable h. the following corollary.
• Could the height be 5 inches?
Explain. No. If the height were Corollary (Integral Zero Theorem)
5 inches, then the width would be If the coefficients of a polynomial function are integers such that a0  1 and an  0,
0 inches. any rational zeros of the function must be factors of an.

Example 1 Identify Possible Zeros

List all of the possible rational zeros of each function.
a. f(x)  2x3
11x2  12x  9
p
If

is a rational zero, then p is a factor of 9 and q is a factor of 2. The possible
q
values of p are 1, 3, and 9. The possible values for q are 1 and 2.
p 1 3 9
So,

 1, 3, 9, 

, 

, and 

.
q 2 2 2

378 Chapter 7 Polynomial Functions

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 7 Resource Masters Graphing Calculator and 5-Minute Check Transparency 7-6
• Study Guide and Intervention, pp. 405–406 Spreadsheet Masters, p. 39 Answer Key Transparencies
• Skills Practice, p. 407
• Practice, p. 408 Technology
• Reading to Learn Mathematics, p. 409 Interactive Chalkboard
• Enrichment, p. 410
 b. f(x)  x3
9x2
x  105
Since the coefficient of x3 is 1, the possible rational zeros must be a factor of the
constant term 105. So, the possible rational zeros are the integers 1, 3, 5,
2 Teach
7, 15, 21, 35, and 105.
IDENTIFY RATIONAL
ZEROS
FIND RATIONAL ZEROS Once you have written the possible rational zeros,
you can test each number using synthetic substitution. In-Class Example Power
Point®

Example 2 Use the Rational Zero Theorem Teaching Tip While discussing
GEOMETRY The volume of a rectangular solid is 675 cubic centimeters. The Example 1, point out that 1 will
width is 4 centimeters less than the height, and the length is 6 centimeters more always be possible rational zeros.
than the height. Find the dimensions of the solid. Also make sure students clearly
Let x  the height, x  4  the width, and x  6  the length. understand that these are just
Study Tip possible zeros. Until each poten-
Write an equation for the volume.
Descartes’ Rule tial zero has been tested by
of Signs x(x  4)(x  6)  675 Formula for volume x cm synthetic substitution, it should
Examine the signs of the x3  2x2  24x  675 Multiply. not be referred to as a zero.
coefficients in the
x
4 cm
equation,  

. x3  2x2  24x  675  0 Subtract 675. 1 List all of the possible
x  6 cm
There is one change of
sign, so there is only one rational zeros of each function.
positive real zero. The leading coefficient is 1, so the possible integer zeros are factors of 675, 1, 3, a. f(x)  3x4  x3  4
5, 9, 15, 25, 27, 45, 75, 135, 225, and 675. Since length can only 1 2 4
be positive, we only need to check positive zeros. From Descartes’ Rule of Signs, 1, 2, 4,   ,   ,  
3 3 3
we also know there is only one positive real zero. Make a table and test possible
real zeros. b. f(x)  x4  7x3  15
p 1 2
24
675 1, 3, 5, 15
1 1 3 21 696
3 1 5 9 702
5 1 7 11 620
9 1 11 75 0
FIND RATIONAL ZEROS
In-Class Examples Power
Point®
One zero is 9. Since there is only one positive real zero, we do not have to test
the other numbers. The other dimensions are 9  4 or 5 centimeters and 9  6 or
15 centimeters. 2 GEOMETRY The volume of a
rectangular solid is 1120 cubic
CHECK Verify that the dimensions are correct. 5  9  15  675 ⻫
feet. The width is 2 feet less
than the height, and the length
You usually do not need to test all of the possible zeros. Once you find a zero, you is 4 feet more than the height.
can try to factor the depressed polynomial to find any other zeros.
Find the dimensions of the
Example 3 Find All Zeros solid.
Find all of the zeros of f(x)  2x4
13x3  23x2
52x  60.
From the corollary to the Fundamental p x ft
 2
13 23
52 60
Theorem of Algebra, we know there are q
exactly 4 complex roots. According to 1 2 11 34 18 20
Descartes’ Rule of Signs, there are 4, 2, or 0
2 2 9 5 42 24
positive real roots and 0 negative real roots. x
2 ft
The possible rational zeros are 1, 2, 3 2 7 2 46 78
3, 4, 5, 6, 10, 12, 15, 20, 30, x  4 ft
5 2 3 8 12 0
1 3 5 15 length: 14 ft, width: 8 ft,
60, 

, 

, 

, and 

. Make a table
2 2 2 2
and test some possible rational zeros. (continued on the next page) height: 10 ft
www.algebra2.com/extra_examples Lesson 7-6 Rational Zero Theorem 379 3 Find all of the zeros of f(x) 
x4  x3  19x2  11x  30.

5,
1, 2, 3
Differentiated Instruction
Logical Organize the students in groups of four or five. Have the students
in each group split the work shown in Example 3 into four or five steps,
depending on the size of their group. Each student then prepares and gives
an explanation to the group of their part of the Example. In particular, stu-
dents should explain any mathematical processes, what the result of their
step is to be, and how the result relates to the next step in the process.

Lesson 7-6 Rational Zero Theorem 379

 Since f(5)  0 , you know that x  5 is a zero. The depressed polynomial is
3 Practice/Apply 2x3  3x2  8x  12.
Factor 2x3  3x2  8x  12.
2x3  3x2  8x  12  0 Write the depressed polynomial.
2x3  8x  3x2  12  0 Regroup terms.

Study Notebook 2x(x2  4)  3(x2  4)  0 Factor by grouping.

(x2  4)(2x  3)  0 Distributive Property
Have students—
• add the definitions/examples of x2 40 or 2x  3  0 Zero Product Property

the vocabulary terms to their x2  4 2x  3

3
Vocabulary Builder worksheets for x  2i x 


2
Chapter 7. 3
There is another real zero at x 

and two imaginary zeros at x  2i and x  2i.
2
• write notes summarizing the 3
The zeros of this function are 5, , 2i and 2i.



Rational Zero Theorem and the 2

Integral Zero Theorem.

• include any other item(s) that they
find helpful in mastering the skills
in this lesson.
Concept Check 1. Explain why it is useful to use the Rational Zero Theorem when finding the
zeros of a polynomial function. Sample answer: You limit the number of possible
solutions.
2. OPEN ENDED Write a polynomial function that has possible rational zeros of
1 3
1, 3, 

, 

. Sample answer: 2x2  x  3
2 2
FIND THE ERROR 3. FIND THE ERROR Lauren and Luis are listing the possible rational zeros of
Suggest students f(x)  4x5  4x4  3x3  2x2 – 5x  6. Luis; Lauren
begin by assuming found numbers in
Lauren Luis q p
that both Lauren and Luis are the form

, not


p q
incorrect. While students may +– 1, +–
1
, +–
1
, +–
1
, 1 1
–+ 1, –+
2
, –+
4
, –+ 2, as Luis did
2 3 6 according to the
spot Lauren’s mistake quickly by
looking at the possible integral +– 2, +–
2
, +– 4, +–
4
3 3
–+ 3, –+
2
, –+
4
, –+ 6 Rational Zero
3 3
Theorem.
zeros she has listed, ask students
not to conclude that Luis must be Who is correct? Explain your reasoning.
correct without actually checking
the possibilities he listed. Guided Practice List all of the possible rational zeros of each function.
4. p(x)  x4  10 1, 2, 5, 10 5. d(x)  6x3  6x2  15x  2
1 1 1 2
1, 2, 

, 

, 

, 


GUIDED PRACTICE KEY Find all of the rational zeros of each function. 2 3 6 3
About the Exercises… Exercises Examples 6. p(x)  x3  5x2  22x  56
4, 2, 7 7. f(x)  x3  x2  34x  56
2,
4, 7
Organization by Objective 4, 5 1 7
6–9, 11 2 8. t(x)  x4  13x2  36 2,
2, 3,
3 9. f(x)  2x3  7x2  8x  28
2, 2,


• Identify Rational Zeros: 2
10 3 2
3 
 1
7
12–17 10. Find all of the zeros of f(x)  6x  5x  9x  2.

,


3 2
3 4
• Find Rational Zeros: 18–33
Application 11. GEOMETRY The volume of the rectangular solid is
Odd/Even Assignments 1430 cubic centimeters. Find the dimensions of the ᐉ  3 cm
Exercises 12–33 are structured solid.
so that students practice the 10 cm  11 cm  13 cm
same concepts whether they ᐉ  1 cm
ᐉ cm
are assigned odd or even
problems. 380 Chapter 7 Polynomial Functions

Assignment Guide
Basic: 13–33 odd, 37, 38, 42–61
Average: 13–33 odd, 34–38,
42–61
Advanced: 12–32 even, 34–36,
39–55 (optional: 56–61)
All: Practice Quiz 2 (1–5)

380 Chapter 7 Polynomial Functions

 NAME ______________________________________________ DATE ____________ PERIOD _____

Practice and Apply Study

7-6 Guide
Study andIntervention
Guide and Intervention,
p. 405 (shown)
Rational Zero Theoremand p. 406

Homework Help List all of the possible rational zeros of each function. Identify Rational Zeros

Lesson 7-6
Rational Zero Let f(x)  a0x n  a1x n  1  …  an  2x 2  an  1x  an represent a polynomial function
For See
12. f(x)  x3  6x  2 1, 2 13. h(x)  x3  8x  6 1, 2, 3, 6
p
Theorem with integral coefficients. If
q
is a rational number in simplest form and is a zero of y  f(x),
then p is a factor of an and q is a factor of a0.
Exercises Examples Corollary (Integral If the coefficients of a polynomial are integers such that a0  1 and an  0, any rational

12–17 1 14. f(x)  3x4  15 15. n(x)  x5  6x3  12x  18 Zero Theorem) zeros of the function must be factors of an.

18–29, 2 Example
16. p(x)  3x3  5x2  11x  3 17. h(x)  9x6  5x3  27
List all of the possible rational zeros of each function.
a. f(x)  3x4
2x2  6x
10
34–41 p
If
q
is a rational root, then p is a factor of 10 and q is a factor of 3. The possible values
30–33 3 for p are 1, 2, 5, and 10. The possible values for q are 1 and 3. So all of the

Find all of the rational zeros of each function. 18.
6,
5, 10
p 1 2 5 10
possible rational zeros are
q
 1, 2, 5, 10, 

, 

, 

, and 

.
3 3 3 3

Extra Practice 18. f(x)  x3  x2  80x  300 19. p(x)  x3  3x  2
1,
1, 2
b. q(x)  x3
10x2  14x
36
Since the coefficient of x3 is 1, the possible rational zeros must be the factors of the
constant term 36. So the possible rational zeros are 1, 2, 3, 4, 6, 9, 12, 18,
See page 843. and 36.
20. h(x)  x4  x2  2 1,
1 21. g(x)  x4  3x3  53x2  9x 0, 9 Exercises
14. 1, 3, 5, 1
22. f(x)  2x5  x4  2x  1

,
1, 1 23. f(x)  x5  6x3  8x 0, 2,
2 List all of the possible rational zeros of each function.

1 5 2 1. f(x)  x3  3x2  x  8 2. g(x)  x5  7x4  3x2  x  20

15, 

, 

24. g(x)  x  3x  x  3x 0, 3
4 3 2 25. p(x)  x4  10x3  33x2  38x  8 1, 2, 4, 8 1, 2, 4, 5, 10, 20
3 3
15. 1, 2, 3, 6, 26. p(x)  x3  3x2  25x  21
7, 1, 3 27. h(x)  6x3  11x2  3x  2 3. h(x)  x4  7x3  4x2  x  49 4. p(x)  2x4  5x3  8x2  3x  5

9, 18 1, 7, 49 1, 5, 
, 

1 5

28. h(x)  10x3  17x2  7x  2 29. g(x)  48x4  52x3  13x  3

2 2
1
16. 1, 

, 3 1 1 1 1 1 1 1 3 5. q(x)  3x4  5x3  10x  12 6. r(x)  4x5  2x  18
3 25.
2,
4 27.

,


,
2 28.


,

, 2 29.


,

,

,

1, 2, 3, 4, 6, 12, 1, 2, 3, 6, 9, 18,

1 1 2 3 2 5 2 3 2 4 1 2

, 
, 

3 3
4
3
1 3 9

, 
, 
, 
, 
, 

2 2 2
1
4
3
4
9
4
17. 1, 

, 

, Find all of the zeros of each function. 30–33. See margin. 7. f(x)  x7  6x5  3x4  x3  4x2  120 8. g(x)  5x6  3x4  5x3  2x2  15
3 9
30. p(x)  6x4  22x3  11x2  38x  40 31. g(x)  5x4  29x3 55x2  28x
1 3
1, 2, 3, 4, 5, 6, 8, 10, 12, 1, 3, 5, 15, 
, 

3, 9, 27 15, 20, 24, 30, 40, 60, 120 5 5

32. h(x)  9x5  94x3  27x2  40x  12 33. p(x)  x5  2x4  12x3  12x2  13x  10 9. h(x)  6x5  3x4  12x3  18x2  9x  21
1 3 7 21
10. p(x)  2x7  3x6  11x5  20x2  11
1 11
1, 3, 7, 21, 
, 
, 
, 
, 1, 11, 
, 

2 2 2 2 2 2
1 7 1 7

, 
, 
, 

FOOD For Exercises 34–36, use the following information. 35. 2,
3  i
3 ; 2 Gl
3

/M
3

NAME
G Hill
6 6

______________________________________________
405 DATE ____________
GlPERIOD
Al _____
b 2
Terri’s Ice Cream Parlor makes gourmet ice cream cones. The volume of each cone is Skills
7-6 Practice,
Practice (Average)
p. 407 and
8 cubic inches. The height is 4 inches more than the radius of the cone’s opening. Practice, p.Theorem
Rational Zero 408 (shown)
List all of the possible rational zeros of each function.
34. Write a polynomial equation that represents the volume of an ice cream cone. 1. h(x)  x3  5x2  2x  12 2. s(x)  x4  8x3  7x  14
1 1 4
Use the formula for the volume of a cone, V 

r2h. V 

r 3 

r 2 1, 2, 3, 4, 6, 12 1, 2, 7, 14
3 3 3 3. f(x)  3x5  5x2  x  6 4. p(x)  3x2  x  7
35. What are the possible values of r? Which of these values are reasonable? 1 2
, , 1, 2, 3, 6
3 3
1 7
, , 1, 7
3 3

36. Find the dimensions of the cone. r  2 in., h  6 in.

5. g(x)  5x3  x2  x  8 6. q(x)  6x5  x3  3
1 2 4 8 1 1 1 3
, , , , 1, 2, 4, 8 , , , , 1, 3
5 5 5 5 6 3 2 2

Find all of the rational zeros of each function.

AUTOMOBILES For Exercises 37 and 38, use 7. q(x)  x3  3x2  6x  8  0
4,
1, 2 8. v(x)  x3  9x2  27x  27 3

the following information. 9. c(x)  x3  x2  8x  12
3, 2 10. f(x)  x4  49x2 0,
7, 7

The length of the cargo space in a sport-utility 11. h(x)  x3  7x2  17x  15 3 12. b(x)  x3  6x  20
2

vehicle is 4 inches greater than the height of the h

13. f(x)  x3  6x2  4x  24 6 14. g(x)  2x3  3x2  4x  4
2
9
space. The width is sixteen inches less than 4
3, 2, 
2 16. z(x)  x4  3x3  5x2  27x  36
1, 4
h
15. h(x)  2x3  7x2  21x  54  0

twice the height. The cargo space has a total ᐉ 17. d(x)  x4  x3  16 no rational zeros 18. n(x)  x4  2x3  3
1
w
Food volume of 55,296 cubic inches. 
2h 19. p(x)  2x4  7x3  4x2  7x  6 20. q(x)  6x4  29x3  40x2  7x  12


The largest ice cream 16 3

1, 1, , 2
3

,

4
37. Write a polynomial function that represents 2 2 3
sundae, weighing
24.91 tons, was made the volume of the cargo space. Find all of the zeros of each function.
21. f(x)  2x4  7x3  2x2  19x  12 22. q(x)  x4  4x3  x2  16x  20
1 
33
 , 
1

33

in Edmonton, Alberta, 38. Find the dimensions of the cargo space.
1,
3, 
4 4

2, 2, 2  i, 2
i

in July 1988.
1  i
3
,
37. V  2h 3
8h 2
64h 38.   36 in., w  48 in., h  32 in. 23. h(x)  x6  8x3

1, 1, 
24. g(x)  x6  1

1
i
3
 1  i
3
2
 1
i
3

Source: The Guinness Book 0, 2,
1  i
3
,
1
i
3
  ,  , 
of Records. AMUSEMENT PARKS For Exercises 39–41, use the following information. 2 2 2
25. TRAVEL The height of a box that Joan is shipping is 3 inches less than the width of the
An amusement park owner wants to add a new wilderness water ride that includes box. The length is 2 inches more than twice the width. The volume of the box is 1540 in3.
What are the dimensions of the box? 22 in. by 10 in. by 7 in.
a mountain that is shaped roughly like a pyramid. Before building the new 26. GEOMETRY The height of a square pyramid is 3 meters shorter than the side of its base.
attraction, engineers must build and test a scale model. 39. V 
1
3
32 1
If the volume of the pyramid is 432 m3, how tall is it? Use the formula V 

Bh. 9 m
3
3 NAME ______________________________________________
408 DATE ____________
Gl PERIOD
Al _____
39. If the height of the scale model is 9 inches less than its length and its base is a Gl
Reading
/M G Hill b 2

7-6 Readingto to Learn

Learn Mathematics
square, write a polynomial function that describes the volume of the model in Mathematics, p. 409 ELL
1 Rational Zero Theorem
terms of its length. Use the formula for the volume of a pyramid, V 

Bh. Pre-Activity How can the Rational Zero Theorem solve problems involving large
3

Lesson 7-6
numbers?
1
40. 6300 


3
3
2 40. If the volume of the model is 6300 cubic inches, write an equation for the situation. Read the introduction to Lesson 7-6 at the top of page 378 in your textbook.

3 Rewrite the polynomial equation w(w  8)(w  5)  2772 in the form

41. What are the dimensions of the scale model?
 30 in., w  30 in., h  21 in.
f(x)  0, where f(x) is a polynomial written in descending powers of x.
w 3  3w 2
40w
2772  0

Reading the Lesson

42. CRITICAL THINKING Suppose k and 2k are zeros of f(x)  x3  4x2  9kx  90. 1. For each of the following polynomial functions, list all the possible values of p, all the
p
possible values of q, and all the possible rational zeros

.
Find k and all three zeros of f(x). k 
3;
3,
6, 5
q
a. f(x)  x3  2x2  11x  12
possible values of p: 1, 2, 3, 4, 6, 12

www.algebra2.com/self_check_quiz Lesson 7-6 Rational Zero Theorem 381 possible values of q: 1

p
possible values of

: 1, 2, 3, 4, 6, 12
q
b. f(x)  2x4  9x3  23x2  81x  45
possible values of p: 1, 3, 5, 9, 15, 45

NAME ______________________________________________ DATE ____________ PERIOD _____ possible values of q: 1, 2

Answers Enrichment,
7-6 Enrichment p. 410 q
p
possible values of

: 1, 3, 5, 9, 15, 45, , , , , , 
1
2
3
2
5
2
9
2
15
2
45
2
2. Explain in your own words how Descartes’ Rule of Signs, the Rational Zero Theorem, and

4
3  i
synthetic division can be used together to find all of the rational zeros of a polynomial

30.
2,  , 
function with integer coefficients.
Infinite Continued Fractions
Sample answer: Use Descartes’ Rule to find the possible numbers of
3 2 Some infinite expressions are actually equal to real
numbers! The infinite continued fraction at the right is
x1
1
1
positive and negative real zeros. Use the Rational Zero Theorem to list all
1
1
possible rational zeros. Use synthetic division to test which of the

5  i
3
one example. 1 numbers on the list of possible rational zeros are actually zeros of the
4 1
1

31.  , 0,  If you use x to stand for the infinite fraction, then the 1… polynomial function. (Descartes’ Rule may help you to limit the
entire denominator of the first fraction on the right is possibilities.)
5 2 also equal to x. This observation leads to the following
equation: Helping You Remember
2
3 
13

1
2 x  1 

32. 3,  ,
 , 
x 3. Some students have trouble remembering which numbers go in the numerators and which
go in the denominators when forming a list of possible rational zeros of a polynomial
3 3 2 Write a decimal for each continued fraction. function. How can you use the linear polynomial equation ax  b  0, where a and b are
nonzero integers, to remember this?
1 1 1 b
1. 1 

2 2. 1  1.5 3. 1  1.666 Sample answer: The solution of the equation is
. The numerator
33.
1,
2, 5, i,
i 1 1 1 a
1 1
1
1
1 b is a factor of the constant term in ax  b. The denominator a is a factor
1
of the leading coefficient in ax  b.

Lesson 7-6 Rational Zero Theorem 381

 43. WRITING IN MATH Answer the question that was posed at the beginning of
4 Assess the lesson. See margin.
How can the Rational Zero Theorem solve problems involving large
numbers?
Open-Ended Assessment Include the following items in your answer:
Speaking Have students explain • the polynomial equation that represents the volume of the compartment, and
how to find all the possible • a list of all reasonable measures of the width of the compartment, assuming
rational zeros of a polynomial that the width is a whole number.
function. Ask them to demonstrate
the technique using a polynomial Standardized 44. Using the Rational Zero Theorem, determine which of the following is a zero of
function of degree 3 or higher Test Practice the function f(x)  12x5  5x3  2x  9. D
3 2
while explaining the process. A 6 B

C 

D 1
8 3
45. OPEN ENDED Write a polynomial with 5, 2, 1, 3, and 4 as roots.
Sample answer: x 5
x 4
27x 3  41x 2  106x
120
Getting Ready for
Lesson 7-7 Maintain Your Skills
PREREQUISITE SKILL Students will Mixed Review Given a function and one of its zeros, find all of the zeros of the function.
perform arithmetic operations on (Lesson 7-5)
functions and find the composi- 46.
6,
3, 5 46. g(x)  x3  4x2  27x  90; 3 47. h(x)  x3  11x  20; 2  i
tion of functions in Lesson 7-7. 47.
4, 2  i, 2
i 48. f(x)  x3  5x2  9x  45; 5 49. g(x)  x3  3x2  41x  203; 7
These skills will rely on students’
5, 3i,
3i
7, 5  2i, 5
2i
ability to correctly perform Given a polynomial and one of its factors, find the remaining factors of the
polynomial. Some factors may not be binomials. (Lesson 7-4)
operations on polynomials. Use
Exercises 56–61 to determine your 50. 20x3  29x2  25x  6; x  2 51. 3x4  21x3  38x2  14x  24; x  3
4x  3, 5x
1 x
4, 3x2  2
students’ familiarity with perform- Simplify. (Lesson 5-5)
ing operations with polynomials. 52. 245 53.  18x3
y2 54. 
16x2 
40x 
25
56.  x3
6
4x 2
57. 4x 2
8x  3

7
5 x
3xy
2 4x
5
Assessment Options 55. GEOMETRY The perimeter of a right triangle is 24 centimeters. Three times
58. x 3  5x 2  x
10 the length of the longer leg minus two times the length of the shorter leg
Practice Quiz 2 The quiz pro- 59. x 5
7x 4  8x 3  exceeds the hypotenuse by 2 centimeters. What are the lengths of all three sides?
vides students with a brief review 106x 2
85x  25 (Lesson 3-5) 6 cm, 8 cm, 10 cm
of the concepts and skills in
Lessons 7-4 through 7-6. Lesson Getting Ready for PREREQUISITE SKILL Simplify.
the Next Lesson (To review operations with polynomials, see Lessons 5-2 and 5-3.)
numbers are given to the right of 33
60. x
9 

56. (x2  7)  (x3  3x2  1) 57. (8x2  3x)  (4x2  5x  3)
the exercises or instruction lines x7
61. x2  x
4  58. (x  2)(x2  3x  5) 59. (x3  3x2  3x  1)(x  5)2
so students can review concepts
not yet mastered. 5


60. (x2  2x  30)
(x  7) 61. (x3  2x2  3x  1)
(x  1)
x1

Answer P ractice Quiz 2 Lessons 7-4 through 7-6

43. The Rational Zero Theorem helps Use synthetic substitution to find f(2) and f(3) for each function. (Lesson 7-4)
factor large numbers by eliminat- 1. f(x)  7x5  25x4  17x3  32x2  10x  22
930,
145
ing some possible zeros because 2. f(x)  3x4  12x3  21x2  30x 0,
180
it is not practical to test all of
3. Write the polynomial equation of degree 4 with leading coefficient 1 that has roots
them using synthetic substitution. at 2, 1, 3, and 4. (Lesson 7-5) x 4
4x 3
7x 2  22x  24  0
Answers should include the
following. Find all of the rational zeros of each function. (Lesson 7-6)
4 3
4. f(x)  5x3  29x2  55x  28

5. g(x)  4x3  16x2  x  24



• The polynomial equation that 5 2
represents the volume of the 382 Chapter 7 Polynomial Functions
compartment is
V  w 3  3w 2
40w.
• Reasonable measures of the
width of the compartment are,
in inches, 1, 2, 3, 4, 6, 7, 9, 12,
14, 18, 21, 22, 28, 33, 36, 42,
44, 63, 66, 77, and 84. The
solution shows that w  14 in.,

 22 in., and d  9 in.

382 Chapter 7 Polynomial Functions

 Operations on Functions Lesson
Notes

• Find the sum, difference, product, and quotient of functions.

• Find the composition of functions.
1 Focus
Vocabulary is it important to combine functions in business?
• composition of functions Carol Coffmon owns a garden store where she
5-Minute Check
sells birdhouses. The revenue from the sale of the Transparency 7-7 Use as a
birdhouses is given by r(x)  125x. The function quiz or review of Lesson 7-6.
for the cost of making the birdhouses is given by
c(x)  65x  5400. Her profit p is the revenue minus Mathematical Background notes
the cost or p  r  c. So the profit function p(x) are available for this lesson on
can be defined as p(x)  (r  c)(x). If you have
two functions, you can form a new function by p. 344D.
performing arithmetic operations on them.
is it important to
combine functions in
ARITHMETIC OPERATIONS Let f(x) and g(x) be any two functions. You can
add, subtract, multiply, and divide functions according to the following rules. business?
Ask students:
Operations with Function
• What is the difference between
Operation Definition Examples if f(x)  x  2, g(x)  3x revenue and profit? Revenue is
Sum (f  g)(x)  f(x)  g(x) (x  2)  3x  4x  2 the amount of money generated by
Difference (f  g)(x)  f(x)  g(x) (x  2)  3x  2x  2 the sales of the birdhouses. Profit
Product (f • g)(x)  f(x) • g(x) (x  2)3x  3x2  6x
is the amount that revenue exceeds
the cost (or expenses) of producing
x2

g
(x) 

f fx
( )
Quotient
, g(x)  0
g(x)



3x
the birdhouses.
• Is profit always a positive value?
Explain. No; profit is negative when
Example 1 Add and Subtract Functions expenses exceed revenue, in which
Given f(x)  x2
3x  1 and g(x)  4x  5, find each function. case it is called loss.
a. ( f  g)(x)
( f  g)(x)  f(x)  g(x)
 (x2  3x  1)  (4x  5)
Addition of functions
f(x)  x2  3x  1 and g(x)  4x  5
2 Teach
 x2  x  6 Simplify.
ARITHMETIC OPERATIONS
b. ( f
g)(x)
( f  g)(x)  f(x)  g(x) Subtraction of functions
In-Class Example Power
Point®
 (x2  3x  1)  (4x  5) f(x)  x2  3x  1 and g(x)  4x  5
 x2  7x  4 Simplify.
1 Given f(x)  3x2  7x and
g(x)  2x2  x  1, find each
Notice that the functions f and g have the same domain of all real numbers. The
function.
functions f  g and f  g also have domains that include all real numbers. For each a. ( f  g)(x) 5x 2  6x
1
new function, the domain consists of the intersection of the domains of f(x) and g(x).
The domain of the quotient function is further restricted by excluded values that b. ( f  g)(x) x 2  8x  1
make the denominator equal to zero.
Lesson 7-7 Operations on Functions 383

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 7 Resource Masters Graphing Calculator and 5-Minute Check Transparency 7-7
• Study Guide and Intervention, pp. 411–412 Spreadsheet Masters, p. 40 Real-World Transparency 7
• Skills Practice, p. 413 School-to-Career Masters, p. 13 Answer Key Transparencies
• Practice, p. 414 Teaching Algebra With Manipulatives
• Reading to Learn Mathematics, p. 415 Masters, pp. 253–255 Technology
• Enrichment, p. 416 Interactive Chalkboard
• Assessment, p. 444

Lesson x-x Lesson Title 383

In-Class Example Power Example 2 Multiply and Divide Functions
Point®
Given f(x)  x2  5x
1 and g(x)  3x
2, find each function.
2 Given f(x)  3x2  2x  1 a. (f • g)(x)
and g(x)  x  4, find each ( f • g)(x)  f(x) • g(x) Product of functions
function.  (x2  5x  1)(3x  2) f(x)  x2  5x  1 and g(x)  3x  2
a. ( f g)(x) 3x 3
14x 2  9x
4  x2(3x  2)  5x(3x  2)  1(3x  2) Distributive Property

3x 2
2x  1  3x3  2x2  15x2  10x  3x  2

 
f Distributive Property
b.
(x)  ,x4  3x3  13x2  13x  2
g x
4 Simplify.

g
f
b.

(x)

COMPOSITION OF 
gf
(x) 
gf((x
x)) Division of functions

FUNCTIONS x2  5x  1 2


, x 

f(x)  x2  5x  1 and g(x)  3x  2
3x  2 3

Intervention
2 2 f
Because x 

makes 3x  2  0,

is excluded from the domain of

(x).
3 3 g 
New Some students
may read f  g
as the word fog. Study Tip COMPOSITION OF FUNCTIONS Functions can also be combined using
composition of functions . In a composition, a function is performed, and then a
Listen for stu- Reading Math second function is performed on the result of the first function. The composition
dents making this verbal error. [f  g](x) and f[g(x)] are
of f and g is denoted by f  g.
both read f of g of x.
Stress that students must learn
to read this correctly because
the correct wording will help Composition of Functions
them understand the meaning. Suppose f and g are functions such that the range of g is a subset of the domain
Lead students to understand of f. Then the composite function f  g can be described by the equation
the similarity in meaning be- [f  g](x)  f[g(x)].
tween f(x) (read “f of x”) and
f  g (read “f of g of x”). Rein-
force this understanding by The composition of functions can be shown by mappings. Suppose f  {(3, 4),
showing how f  g can also be (2, 3), (5, 0)} and g  {(3, 5), (4, 3), (0, 2)}. The composition of these functions is
shown below.
written as f [g(x)]. You can also
relate f [g(x)] to an expression f°g g°f
containing nested parentheses,
such as (1  (3 5(4))) in which x 3 4 0 domain of g x 3 2
5 domain of f
the expressions in parentheses
are evaluated from the
range of g range of f
innermost parentheses to the g (x )
5 3 2
domain of f
f (x ) 4 3 0
domain of g
outermost.

f [g (x )] 0 4 3 range of f g [f (x )] 3
5 2 range of g

f ° g  {(3, 0), (4, 4), (0, 3)} g ° f  {(3, 3), (2,
5), (
5, 2)}

The composition of two functions may not exist. Given two functions f and g,
[ f  g](x) is defined only if the range of g(x) is a subset of the domain of f(x). Similarly,
[ g  f ](x) is defined only if the range of f(x) is a subset of the domain of g(x).
384 Chapter 7 Polynomial Functions

384 Chapter 7 Polynomial Functions

 Example 3 Evaluate Composition of Relations In-Class Examples Power
Point®
If f(x)  {(7, 8), (5, 3), (9, 8), (11, 4)} and g(x)  {(5, 7), (3, 5), (7, 9), (9, 11)},
find f ⴗ g and g ⴗ f. 3 If f(x)  {(2, 6), (9, 4), (7, 7),
To find f  g, evaluate g(x) first. Then use the range of g as the domain of f and (0, 1)} and g(x)  {(7, 0),
evaluate f(x). (1, 7), (4, 9), (8, 2)}, find f  g
f [g(5)]  f(7) or 8 g(5)  7 and g  f. f
g  {(7,
1),
f [g(3)]  f(5) or 3 g(3)  5 (
1, 7), (4, 4), (8, 6)};
f [g(7)]  f(9) or 8 g(7)  9 g
f  {(9, 9), (7, 0), (0, 7)}
f [g(9)]  f(11) or 4 g(9)  11
4
f  g  {(5, 8), (3, 3), (7, 8), (9, 4)}
a. Find [f  g](x) and [g  f ](x)
To find g  f, evaluate f(x) first. Then use the range of f as the domain of g and for f(x)  3x2  x  4 and
evaluate g(x).
g(x)  2x  1.
g[ f(7)]  g(8) g(8) is undefined. [f
g](x)  12x2
14x  8;
g[ f(5)]  g(3) or 5 f(5)  3 [g
f ](x)  6x2
2x  7
g[ f(9)]  g(8) g(8) is undefined.
b. Evaluate [f  g](x) and
g[ f(11)]  g(4) g(4) is undefined.
[g  f ](x) for x  2.
Since 8 and 4 are not in the domain of g, g  f is undefined for x  7, x  9, and [f
g](
2)  84;
x  11. However, g[ f(5)]  5 so g  f  {(5, 5)}. [g
f ](
2)  35

Notice that in most instances f  g  g  f. Therefore, the order in which you

compose two functions is very important.

Example 4 Simplify Composition of Functions

a. Find [f ⴗ g](x) and [g ⴗ f](x) for f(x)  x  3 and g(x)  x2  x
1.

[ f  g](x)  f [g(x)] Composition of functions
 f(x2  x  1) Replace g(x) with x2  x  1.
 (x2  x  1)  3 Substitute x2  x  1 for x in f(x).
 x2  x  2 Simplify.

[g  f ](x)  g[ f(x)] Composition of functions

 g(x  3) Replace f(x) with x  3.
 (x  3)2  (x  3)  1 Substitute x  3 for x in g(x).
 x2  6x  9  x  3  1 Evaluate (x  3)2.
 x2  7x  11 Simplify.

So, [ f  g](x)  x2  x  2 and [g  f ](x)  x2  7x  11.

b. Evaluate [f ⴗ g](x) and [g ⴗ f](x) for x  2.

[ f  g](x)  x2  x  2 Function from part a
[ f  g](2)  (2)2  2  2 Replace x with 2.
8 Simplify.

[ g  f ](x)   7x  11
x2 Function from part a
[ g  f ](2)  (2)2  7(2)  11 Replace x with 2.
 29 Simplify.

So, [ f  g](2)  8 and [g  f ](2)  29.

www.algebra2.com/extra_examples Lesson 7-7 Operations on Functions 385

Differentiated Instruction
Naturalist Invite students to think of events in nature whose occurrence
students can relate to as being similar to the order in which the compo-
sition of functions must be carried out. Students interested in science
might think of the stages of metamorphosis in insects, during which a
larva changes to a pupa and finally to an adult. Each stage is dependent
on the one before and the order of the stages is fixed.

Lesson 7-7 Operations on Functions 385

In-Class Example Power Example 5 Use Composition of Functions
Point®
TAXES Tyrone Davis has $180 deducted from every paycheck for retirement.
5 TAXES Tracie Long has $100 He can have these deductions taken before taxes are applied, which reduces his
taxable income. His federal income tax rate is 18%. If Tyrone earns $2200 every
deducted from every paycheck pay period, find the difference in his net income if he has the retirement
for retirement. She can have Study Tip deduction taken before taxes or after taxes.
this deduction taken before Combining
state taxes are applied, which Functions Explore Let x  Tyrone’s income per paycheck, r(x)  his income after the
By combining functions, deduction for retirement, and t(x)  his income after the deduction for
reduces her taxable income. you can make the federal income tax.
Her state income tax rate is evaluation of the functions
4%. If Tracie earns $1500 every more efficient. Plan Write equations for r(x) and t(x).
$180 is deducted from every paycheck for retirement: r(x)  x  180.
pay period, find the differ-
ence in her net income if she Tyrone’s tax rate is 18%: t(x)  x  0.18x.
has the retirement deduction Solve If Tyrone has his retirement deducted before taxes, then his net income
taken before or after state is represented by [t  r](2200).
taxes. Her net pay is $4 more by [t  r](2200)  t(2200  180) Replace x with 2200 in r(x)  x  180.
having her retirement deduction  t(2020)
taken before state taxes.  2020  0.18(2020) Replace x with 2020 in t(x)  x  0.18x.
 1656.40
If Tyrone has his retirement deducted after taxes, then his net income is
3 Practice/Apply represented by [r  t](2200).
[r  t](2200)  r[2200  0.18(2200)] Replace x with 2200 in t(x)  x  0.18x.
 r(1804)
 1804  180 Replace x with 1804 in r(x)  x  180.

Study Notebook  1624

Have students— [t  r](2200)  1656.40 and [r  t](2200)  1624. The difference is

$1656.40  $1624 or $32.40. So, his net pay is $32.40 more by having his
• add the definitions/examples of retirement deducted before taxes.
the vocabulary terms to their Examine The answer makes sense. Since the taxes are being applied to a smaller
Vocabulary Builder worksheets for amount, less taxes will be deducted from his paycheck.
Chapter 7.
• include any other item(s) that they
find helpful in mastering the skills
Concept Check 1. Determine whether the following statement is always, sometimes, or never true.
in this lesson. Support your answer with an example.
1. Sometimes; sample Given two functions f and g, f  g  g  f.
answer: If f(x)  x

2, g(x)  x  8, then 2. OPEN ENDED Write a set of ordered pairs for functions f and g, given that
f ⴗ g  x  6 and g ⴗ f f  g  {(4, 3), (1, 9), (2, 7)}.
 x  6. 3. FIND THE ERROR Danette and Marquan are finding [g  f](3) for
2. Sample answer: f(x)  x2  4x  5 and g(x)  x  7.
FIND THE ERROR g(x)  {(
2, 1), (
1,
Point out that Danet te Marquan
2), (4, 3)}, f(x)  {(1,
the order in which the 7), (2, 9), (3, 3)} [g  f](3) = g[(3) 2 + 4(3) + 5] [g  f](3) = f(3 – 7)
functions are applied is related to = g(26) = f(–4)
the proximity of the function = 26 - 7 = (–4) 2 + 4(–4) + 5
name to the variable x. Since f is = 19 =5
closer to x here, it is the first
function to be evaluated. Invite Who is correct? Explain your reasoning. See margin.
students to suggest other ways 386 Chapter 7 Polynomial Functions
they can avoid confusing f  g
with g  f.
Answer
3. Danette; [g
f ](x)  g[f(x)]
means to evaluate the f function
first and then the g function.
Marquan evaluated the functions
in the wrong order.

386 Chapter 7 Polynomial Functions

 g
f
Guided Practice Find (f  g)(x), (f
g)(x), (f • g)(x), and

(x) for each f(x) and g(x). 4–5. See margin.
GUIDED PRACTICE KEY 4. f(x)  3x  4 5. f(x)  x2  3 About the Exercises…
g(x)  5  x g(x)  x  4
Exercises Examples Organization by Objective
4–7 1, 2 For each set of ordered pairs, find f ⴗ g and g ⴗ f, if they exist. • Arithmetic Operations:
8, 9 3 17–22
10–14 4 6. f  {(1, 9), (4, 7)} 7. f  {(0, 7), (1, 2), (2, 1)}
15, 16 5 g  {(5, 4), (7, 12), (4, 1)} g  {(1, 10), (2, 0)} • Composition of Functions:
{(
5, 7), (4, 9)}; {(4, 12)} {(2,
7)}; {(1, 0), (2, 10)} 23–46
Find [g ⴗ h](x) and [h ⴗ g](x).
8. g(x)  2x 9. g(x)  x  5
Odd/Even Assignments
h(x)  3x  4 6x
8; 6x
4 h(x)  x2  6 x 2  11; x 2  10x  31 Exercises 17–46 are structured
so that students practice the
If f(x)  3x , g(x)  x  7, and h(x)  x2, find each value. same concepts whether they
10. f [g(3)] 30 11. g[h(2)] 11 12. h[h(1)] 1 are assigned odd or even
problems.
Application SHOPPING For Exercises 13–15, use the following information.
14. $32.50; price of
Mai-Lin is shopping for computer software. She finds a CD-ROM program that costs
$49.99, but is on sale at a 25% discount. She also has a $5 coupon she can use on the
Assignment Guide
CD when 25% dis- product. Basic: 17–45 odd, 47, 48, 56–81
count is taken and
then the coupon is 13. Express the price of the CD after the discount and the price of the CD after Average: 17–45 odd, 47, 48,
the coupon using function notation. Let x represent the price of the CD, p(x)
subtracted
represent the price after the 25% discount, and c(x) represent the price after
55–81
15. $33.75; price of the coupon. p(x) 
3
x ; c(x)  x
5 Advanced: 18–46 even, 49–54,
CD when coupon is 4
subtracted and then 14. Find c[p(x)] and explain what this value represents. 56–75 (optional: 76–81)
25% discount is taken
15. Find p[c(x)] and explain what this value represents.
16. Which method results in the lower sale price? Explain your reasoning.
See margin. Answers
4. 4x  9; 2x
1; 3x2  19x  20;
Practice and Apply 3x  4
 , x 
5
x5
g
f
Homework Help Find (f  g)(x), (f
g)(x), (f • g)(x), and

(x) for each f(x) and g(x). 5. x 2  x
1; x 2
x  7;
For See x2  3
Exercises Examples 17. f(x)  x  9 18. f(x)  2x  3 19. f(x)  2x2 x 3
4x 2  3x
12;  , x  4
x
4
17–22 1, 2 g(x)  x  9 g(x)  4x  9 g(x)  8  x
23–28 3 16. Discount first, then coupon;
29–46 4 20. f(x)  x2  6x  9 21. f(x)  x2  1 22. f(x)  x2  x  6
47–55 5 g(x)  2x  6 x x3 sample answer: 25% of 49.99 is
g(x) 

g(x) 


x1 x2 greater than 25% of 44.99.
17–22. See margin.
Extra Practice For each set of ordered pairs, find f ⴗ g and g ⴗ f if they exist. x9
See page 844. 17. 2x; 18; x2
81;  , x  9
23. f  {(1, 1), (0, 3)} 24. f  {(1, 2), (3, 4), (5, 4)} x
9
23. {(1,
3), (
3, 1),
(2, 1)}; {(1, 0), (0, 1)} g  {(1, 0), (3, 1), (2, 1)} g  {(2, 5), (4, 3)} 18. 6x  6;
2x
12; 8x2  6x
27;
2x
3 9
24. {(2, 4), (4, 4)}; 25. f  {(3, 8), (4, 0), (6, 3), (7, 1)} 26. f  {(4, 5), (6, 5), (8, 12), (10, 12)} , x 

{(1, 5), (3, 3), (5, 3)} g  {(0, 4), (8, 6), (3, 6), (1, 8)} g  {4, 6), (2, 4), (6, 8), (8, 10)} 4x  9 4
25. {(0, 0), (8, 3), 19. 2x 2
x  8; 2x 2  x
8;
27. f  {(2, 5), (3, 9), (4, 1)} 28. f  {(7, 0), (5, 3), (8, 3), (9, 2)}
(3, 3)}; {(3, 6), (4, 4), 2x 2
g  {(5, 4), (8, 3), (2, 2)} g  {(2, 5), (1, 0), (2, 9), (3, 6)}
2x 3  16x 2;  , x  8
(6, 6), (7, 8)} {(5, 1), (8, 9)}; {(2,
4)} {(2, 3), (2, 2)}; {(
5, 6), (8, 6), (
9,
5)} 8
x
26. {(4, 5), (2, 5), Find [g ⴗ h](x) and [h ⴗ g](x). 20. x 2  8x  15; x 2  4x  3;
(6, 12), (8, 12)}; does 29. g(x)  4x 8x
4; 30. g(x)  5x 15x
5; 31. g(x)  x  2 x3
not exist 2x 3  18x2  54x  54;  ,
h(x)  2x  1 8x
1 h(x)  3x  1 15x  1 h(x)  x2 2
31. x2  2; x2  4x  4 x 
3
34. 2x2
5x  9; 2x2 32. g(x)  x 2 4 33. g(x)  2x 34. g(x)  x  1
x3  x2
1

x5 h(x) 3x h(x)  x3  x2  x  1 h(x)  2x2  5x  8 21.  , x 
1;
32. 3x2
4; 3x2
24x  48 33. 2x3  2x2  2x  2; 8x3  4x2  2x  1 x1
www.algebra2.com/self_check_quiz Lesson 7-7 Operations on Functions 387
x 3  x 2
2x
1
 , x 
1; x2
x,
x1
x3  x2
x
1
x 
1;  , x  0
x
x 3  x 2
7x
15
22.  , x 
2;
x2
x  x
9x
9
3 2
 , x 
2;
x2
x2
6x  9, x 
2 ;
x2  4x  4, x 
2, 3

Lesson 7-7 Operations on Functions 387

 Study
NAME ______________________________________________ DATE

Guide andIntervention
Intervention,
____________ PERIOD _____
If f(x)  4x, g(x)  2x
1, and h(x)  x2  1, find each value.
7-7 Study Guide and
p. 411 (shown)
Operations and p. 412
on Functions 35. f [ g(1)]
12 36. h[ g(4)] 50 37. g[ f (5)] 39
Arithmetic Operations
Sum (f  g)(x)  f(x)  g(x)
38. f [h(4)] 68 39. g[ g(7)] 25 40. f [ f(3)]
48
Difference (f  g)(x)  f(x)  g(x)
1
Operations with Functions Product
Quotient
(f g)(x)  f(x) g(x)


gf (x) 

f(x)
g(x)
, g(x)  0
1
41. h f


4    2  1
42. g h 


2  1


2
43. [ g  ( f  h)](3) 79
  44. [ f  (h  g)](3) 104 45. [h  (g  f )](2) 226 46. [ f  (g  h)](2) 36
Example f
Find (f  g)(x), (f
g)(x), (f  g)(x), and 
g
(x) for f(x)  x2  3x
4

Lesson 7-7
and g(x)  3x
2.
(f  g)(x)  f(x)  g(x) Addition of functions
 (x2  3x  4)  (3x  2) f(x)  x 2  3x  4, g(x)  3x  2
 x2  6x  6 Simplify.
(f  g)(x)  f(x)  g(x) Subtraction of functions
 (x2  3x  4)  (3x  2) f(x)  x 2  3x  4, g(x)  3x  2
x 2
2 Simplify. POPULATION GROWTH For Exercises 47 and 48, use the following information.
(f g)(x)  f(x) g(x)
 (x2  3x  4)(3x  2)
Multiplication of functions
f(x)  x 2  3x  4, g(x)  3x  2 From 1990 to 1999, the number of births b(x) in the U.S. can be modeled by the
 x2(3x  2)  3x(3x  2)  4(3x  2)
 3x  2x  9x  6x  12x  8
3 2 2
Distributive Property
Distributive Property
function b(x)  27x  4103, and the number of deaths d(x) can be modeled by the
 3x3  7x2  18x  8

gf (x) 

f(x)
Simplify.

Division of functions
function d(x)  23x  2164, where x is the number of years since 1990 and b(x) and
g(x)
x2  3x  4


, x 


2
f(x)  x 2  3x  4 and g(x)  3x  2
d(x) are in thousands.
3x  2 3

Exercises 47. The net increase in population P is the number of births per year minus the
Find ( f  g)(x), (f
g)(x), (f  g)(x), and 
g  f
(x) for each f(x) and g(x). number of deaths per year or P  b  d. Write an expression that can be used
1. f(x)  8x  3; g(x)  4x  5
12x  2; 4x
8; 32x 2  28x
15;
2. f(x)  x2  x  6; g(x)  x  2
x 2  2x
8; x 2
4;
to model the population increase in the U.S. from 1990 to 1999 in function
8x
3 5
, x 

4x  5 4
x 3
x 2
8x  12; x  3, x  2 notation. P(x) 
50x  1939
3. f(x)  3x2  x  5; g(x)  2x  3 4. f(x)  2x  1; g(x)  3x2  11x  4
Shopping 48. Assume that births and deaths continue at the same rates. Estimate the net
3x 2  x  2; 3x 2
3x  8; 3x 2  13x
5;
3x 2
9x  3; Americans spent over $500 increase in population in 2010. 939,000
6x 3
11x 2  13x
15; 6x 3  19x 2
19x  4;
3x2
x  5
 , x  
3 2x
1
 , x   ,
4
1 million on inline skates
2x
3 2 (3x
1)(x  4) 3
5. f(x)  x2  1; g(x) 

1 and equipment in 2000.
x1
1 1 Source: National Sporting Goods
x 2
1   ; x 2
1
 ; x
1; x 3  x 2
x
1, x 
1
x1 x1 Association SHOPPING For Exercises 49–51, use the following information.
Gl NAME
/M G ______________________________________________
Hill 411 DATE ____________
GlPERIOD
Al _____
b 2 Liluye wants to buy a pair of inline skates that are on sale for 30% off the original
Skills
7-7 Practice,
Practice (Average)
p. 413 and price of $149. The sales tax is 5.75%.
Practice,
Operationsp. 414 (shown)
on Functions
f
Find ( f  g)(x), (f
g)(x), (f  g)(x), and  (x) for each f(x) and g(x).
49. Express the price of the inline skates after the discount and the price of the
g

1. f(x)  2x  1 2. f(x)  8x2 3. f(x)  x2  7x  12

inline skates after the sales tax using function notation. Let x represent the price
1
g(x)  x  3 g(x) 
2
x
g(x)  x2  9
of the inline skates, p(x) represent the price after the 30% discount, and s(x)
8x 4  1
3x
2; x  4; 
x2
8x 4
1
, x  0; 2x 2  7x  3; 7x  21;
represent the price after the sales tax. p (x)  0.70x; s(x)  1.0575x
2x 2
5x
3;  , x  0; x 4  7x 3  3x 2
63x
108;
x2
2x  1 x4
, x  3
x
3
8, x  0; 8x 4, x  0 , x  3
x
3 50. s[p(x)]; The 30% 50. Which composition of functions represents the price of the inline skates, p[s(x)]
For each set of ordered pairs, find f
g and g
f if they exist.
4. f  {(9, 1), (1, 0), (3, 4)} 5. f  {(4, 3), (0, 2), (1, 2)}
would be taken off or s[p(x)]? Explain your reasoning.
g  {(0, 9), (1, 3), (4, 1)}
{(0,
1), (
1, 4), (4, 0)};
g  {(2, 0), (3, 1)}
{(
2,
2), (3,
2)};
first, and then the
{(
9, 3), (
1,
9), (3,
1)} {(
4, 1), (0, 0), (1, 0)} sales tax would be 51. How much will Liluye pay for the inline skates? $110.30
6. f  {(4, 5), (0, 3), (1, 6)}
g  {(6, 1), (5, 0), (3, 4)}
7. f  {(0, 3), (1, 3), (6, 8)}
g  {(8, 2), (3, 0), (3, 1)} calculated on this
{(6, 6), (
5, 3), (3,
5)};
{(
4, 0), (0,
4), (1, 1)}
does not exist;
{(0, 0), (1, 0), (6, 2)} price.
Find [ g
h](x) and [h
g](x). TEMPERATURE For Exercises 52–54, use the following information.
8. g(x)  3x
h(x)  x  4
9. g(x)  8x
h(x)  2x  3
10. g(x)  x  6
h(x)  3x2 3x 2  6;
There are three temperature scales: Fahrenheit (°F), Celsius (°C), and Kelvin (K).
3x
12; 3x
4
16x
24;
16x  3 3x 2  36x  108 The function K(C)  C  273 can be used to convert Celsius temperatures to Kelvin.
11. g(x)  x  3 12. g(x)  2x 13. g(x)  x  2
5
h(x)  2x2
2x 2  3;
h(x)  x2  3x  2

2x 2
6x
4;
h(x)  3x2  1
3x 2
1; The function C(F) 

(F  32) can be used to convert Fahrenheit temperatures to
2x 2  12x  18 4x 2
6x  2 3x 2
12x  13
9
Celsius.
If f(x)  x2, g(x)  5x, and h(x)  x  4, find each value.
14. f[ g(1)] 25 15. g[h(2)] 10 16. h[f(4)] 20 52. Write a composition of functions that could be used to convert Fahrenheit
17. f[h(9)] 25 18. h[ g(3)]
11 19. g[f(8)] 320
temperatures to Kelvin. [K ⴗ C](F) 
5
(F
32)  273
21. [f  (h  g)](1) 1 22. [f  (g  h)](4) 1600
20. h[f(20)] 404
9
23. BUSINESS The function f(x)  1000  0.01x2
models the manufacturing cost per item
when x items are produced, and g(x)  150  0.001x2 models the service cost per item. 53. Find the temperature in Kelvin for the boiling point of water and the freezing
Write a function C(x) for the total manufacturing and service cost per item.
C(x)  1150
0.011x 2
n f
point of water if water boils at 212°F and freezes at 32°F. 373 K; 272 K
24. MEASUREMENT The formula f 

converts inches n to feet f, and m 

converts
12 5280
feet to miles m. Write a composition of functions that converts inches to miles.
[m
f ]n  
n 54. While performing an experiment, Kimi found the temperature of a solution at
63,360
NAME ______________________________________________ DATE ____________
different intervals. She needs to record the change in temperature in degrees
Gl /M G Hill 414 Gl PERIOD
Al _____
b 2
Reading
7-7 Readingto
to Learn
Learn Mathematics
Kelvin, but only has a thermometer with a Fahrenheit scale. What will she record
Mathematics, p. 415 ELL when the temperature of the solution goes from 158°F to 256°F? 309.67 K
Operations on Functions
Pre-Activity Why is it important to combine functions in business?
Read the introduction to Lesson 7-7 at the top of page 383 in your textbook.
Describe two ways to calculate Ms. Coffmon’s profit from the sale of
50 birdhouses. (Do not actually calculate her profit.) Sample answer:
1. Find the revenue by substituting 50 for x in the expression 55. FINANCE Kachina pays $50 each month on a credit card that charges 1.6%
125x. Next, find the cost by substituting 50 for x in the
expression 65x  5400. Finally, subtract the cost from the
revenue to find the profit. 2. Form the profit function
interest monthly. She has a balance of $700. The balance at the beginning of the
p(x)  r(x)
c(x)  125x
(65x  5400)  60x
5400.
Substitute 50 for x in the expression 60x
5400.
nth month is given by f(n)  f(n  1)  0.016 f(n  1)  50. Find the balance at
the beginning of the first five months. No additional charges are made on the
Reading the Lesson
1. Determine whether each statement is true or false. (Remember that true means
card. (Hint: f(1)  700) $700, $661.20, $621.78, $581.73, $541.04
always true.)

a. If f and g are polynomial functions, then f  g is a polynomial function. true 388 Chapter 7 Polynomial Functions
f
b. If f and g are polynomial functions, then

g
is a polynomial function. false

c. If f and g are polynomial functions, the domain of the function f g is the set of all
real numbers. true
f
d. If f(x)  3x  2 and g(x)  x  4, the domain of the function

g
is the set of all real
NAME ______________________________________________ DATE ____________ PERIOD _____
numbers. false

e. If f and g are polynomial functions, then (f  g)(x)  (g  f)(x). false

Enrichment,
7-7 Enrichment p. 416
f. If f and g are polynomial functions, then (f g)(x)  (g f)(x) true

2. Let f(x)  2x  5 and g(x)  x2  1. Relative Maximum Values

a. Explain in words how you would find (f  g)(3). (Do not actually do any calculations.) The graph of f (x)  x3  6x  9 shows a f(x)
Sample answer: Square
3 and add 1. Take the number you get, relative maximum value somewhere
x f (x)

multiply it by 2, and subtract 5. O x 2 5

between f (2) and f (1). You can obtain –4 –2 2 4
a closer approximation by comparing –4 1.5 3.375
b. Explain in words how you would find (g  f)(3). (Do not actually do any
values such as those shown in the table. 1.4 3.344
calculations.) Sample answer: Multiply
3 by 2 and subtract 5. Take the –8
number you get, square it, and add 1. 1.3 3.397
To the nearest tenth a relative maximum –12
1 4
value for f (x) is 3.3.
–16
f(x)  x3
6x
9
Helping You Remember
–20
3. Some students have trouble remembering the correct order in which to apply the two
original functions when evaluating a composite function. Write three sentences, each of
which explains how to do this in a slightly different way. (Hint: Use the word closest in
the first sentence, the words inside and outside in the second, and the words left and Using a calculator to find points, graph each function. To the nearest
right in the third.) Sample answer: 1. The function that is written closest to tenth, find a relative maximum value of the function.
the variable is applied first. 2. Work from the inside to the outside. 1. f (x)  x(x2  3) rel. max. of 2.0 2. f (x)  x3  3x  3 rel. max. of
1.0
3. Work from right to left.
f(x) f(x)

388 Chapter 7 Polynomial Functions

 56. CRITICAL THINKING If f(0)  4 and f(x  1)  3f(x)  2, find f(4). 244

57. WRITING IN MATH Answer the question that was posed at the beginning of
the lesson. See margin.
4 Assess
Why is it important to combine functions in business? Open-Ended Assessment
Include the following in your answer: Modeling In some courses, a
• a description of how to write a new function that represents the profit, using function f(x) is modeled by a
the revenue and cost functions, and “machine” that accepts values
• an explanation of the benefits of combining two functions into one function. for x as inputs and then outputs
values for f(x). Using this model,
Standardized 58. If h(x)  7x  5 and g[h(x)]  2x  3, then g(x)  A
Test Practice ask students to explain how the
2x  31
A

. B 5x  8. composition of two functions
7
2x  26 could be modeled by two such
C 5x  8. D

.
7
“machines” linked together. Also
59. If f(x)  4x4  5x3  3x2  14x  31 and g(x)  7x3  4x2  5x  42, ask them to use the model to
then ( f  g)(x)  C explain how the composition of
A 4x4  12x3  7x2  9x  11. B 4x4  2x3  7x2  19x  11. two functions could be undefined
C 4x4  2x3  x2  19x  73. D 3x4  2x3  7x2  19x  73. for some initial input values.

Maintain Your Skills

Getting Ready for
Lesson 7-8
Mixed Review List all of the possible rational zeros of each function. (Lesson 7-6) PREREQUISITE SKILL In Lesson
1 1
61. 1, 

, 

, 60. r(x)  x2  6x  8 61. f(x)  4x3  2x2  6 62. g(x)  9x2  1
1 1 7-8, students will find the inverse
2 4 1, 2, 4, 8 1, 

, 


3 3 3 9
2, 3, 

, 

, 6 Write a polynomial function of least degree with integral coefficients that has the of a function. One method used
2 4 given zeros. (Lesson 7-5) will involve students solving an
63. x 3
4x 2
17x  63. 5, 3, 4 64. 3, 2, 8
1 2
65. 1,

,

equation for a variable. Use
60 2 3
66. 6, 2i 67. 3, 3  2i 68. 5, 2, 1  i Exercises 76–81 to determine
64. x 3
3x 2
34x
x 3
6x 2  4x
24 x 3
9x 2  31x
39 your students’ familiarity with
48 69. ELECTRONICS There are three basic things to be considered in an electrical solving equations for a variable.
65. 6x 3
13x 2  9x circuit: the flow of the electrical current I, the resistance to the flow Z called

2 impedance, and electromotive force E called voltage. These quantities are related
in the formula E  I • Z. The current of a circuit is to be 35  40j amperes. Assessment Options
68. x 4  x 3
14x 2  Electrical engineers use the letter j to represent the imaginary unit. Find the
26x
20 impedance of the circuit if the voltage is to be 430  330j volts. (Lesson 5-9) 10  2j Quiz (Lessons 7-6 and 7-7) is
available on p. 444 of the Chapter 7
Find the inverse of each matrix, if it exists. (Lesson 4-7) Resource Masters.

6
2
70. 87 65

12

75 8 71. 11 23 
13 1
Answer
72. 86 43 does not exist 73. 43  



2 1 1 2
1 2 3 4
57. Answers should include the
2
3 5 16 3
2
2 1 5 2
  75. 
6 2
74. does not exist

following.
9 3
• Using the revenue and cost
Getting Ready for PREREQUISITE SKILL Solve each equation or formula for the specified variable. functions, a new function that
the Next Lesson (To review solving equations for a variable, see Lesson 1-3.) represents the profit is
6  3y 1
4x2
76. 2x  3y  6, for x x 

77. 4x2  5xy  2  3, for y y 

p(x)  r(c(x)).
2
5x

2 I
78. 3x  7xy  2, for x x 

79. I  prt, for t t 

• The benefit of combining two
3  7y pr
5 9 Mm 2
Fr
functions into one function is
80. C 

(F  32), for F F 

C  32 81. F  G

2 , for m m 

9 5 r GM that there are fewer steps to
Lesson 7-7 Operations on Functions 389 compute and it is less confusing
to the general population of
people reading the formulas.

Lesson 7-7 Operations on Functions 389

 Lesson Inverse Functions and
Notes Relations
• Find the inverse of a function or relation.

1 Focus • Determine whether two functions or relations are inverses.

Vocabulary are inverse functions related to

5-Minute Check • inverse relation
measurement conversions?
Transparency 7-8 Use as a • inverse function Most scientific formulas involve measurements given in SI (International System)
quiz or review of Lesson 7-7. • one-to-one units. The SI units for speed are meters per second. However, the United States
uses customary measurements such as miles per hour. To convert x miles per
Mathematical Background notes hour to an approximate equivalent in meters per second, you can evaluate
are available for this lesson on x miles 1600 meters 1 hour 4
f(x) 
•

•

or f(x) 

x. To convert x meters per
p. 344D. 1 hour 1 mile 3600 seconds 9
second to an approximate equivalent in miles per hour, you can evaluate
are inverse functions
related to measurement x meters 3600 seconds 1 mile 9
g(x) 

•

•

or g(x) 

x.
1 second 1 hour 1600 meters 4
conversions?
Notice that f(x) multiplies a number by 4 and divides it by 9. The function
Ask students: g(x) does the inverse operation of f(x). It divides a number by 4 and
4 9
• When would you need to con- multiplies it by 9. The functions f(x) 

x and g(x) 

x are inverses.
9 4
vert from SI units to customary
units? Sample answer: reading a FIND INVERSES Recall that a relation is a set of ordered pairs. The inverse
recipe printed in a cookbook that relation is the set of ordered pairs obtained by reversing the coordinates of each
was published in Canada original ordered pair. The domain of a relation becomes the range of the inverse,
and the range of a relation becomes the domain of the inverse.
• When would you need to con-
vert between units within the Inverse Relations
customary system? Sample • Words Two relations are inverse relations if and only if whenever one
answer: You might change pounds relation contains the element (a, b), the other relation contains the
element (b, a).
to ounces in order to estimate the
cost per ounce of different grocery • Example Q  {(1, 2), (3, 4), (5, 6)} S  {(2, 1), (4, 3), (6, 5)}
Q and S are inverse relations.
items.
• When you are driving a car at
a speed of 55 miles per hour, Example 1 Find an Inverse Relation
GEOMETRY The ordered pairs of the relation y
how many meters would you
guess you are traveling each {(2, 1), (5, 1), (2,
4)} are the coordinates of the vertices
of a right triangle. Find the inverse of this relation
second? Sample answer: 20 m/s and determine whether the resulting ordered pairs yx

• What is the calculated value are also the vertices of a right triangle.
O x
for f(55)? about 24.4 m/s To find the inverse of this relation, reverse the
coordinates of the ordered pairs.
The inverse of the relation is {(1, 2), (1, 5), (4, 2)}.
Plotting the points shows that the ordered pairs also
describe the vertices of a right triangle. Notice that the
graphs of the relation and the inverse relation are
reflections over the graph of y  x.

390 Chapter 7 Polynomial Functions

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 7 Resource Masters School-to-Career Masters, p. 14 5-Minute Check Transparency 7-8
• Study Guide and Intervention, pp. 417–418 Teaching Algebra With Manipulatives Answer Key Transparencies
• Skills Practice, p. 419 Masters, pp. 256–257, 258
• Practice, p. 420 Technology
• Reading to Learn Mathematics, p. 421 Interactive Chalkboard
• Enrichment, p. 422
Study Tip The ordered pairs of inverse functions are also related. We can write the inverse
Reading Math
f – 1 is read f inverse or the
of function f(x) as f 1(x).
2 Teach
inverse of f. Note that 1 Property of Inverse Functions
is not an exponent. FIND INVERSES
Suppose f and f 1 are inverse functions. Then, f(a)  b if and only if f 1(b)  a.
In-Class Examples Power
Point®
Let’s look at the inverse functions f(x)  x  2 and f1(x)  x  2.
1 GEOMETRY The ordered
Evaluate f(5). Now, evaluate f1(7). pairs of the relation {(1, 3),
f(x)  x  2 f1(x)  x  2 (6, 3), (6, 0), (1, 0)} are the
f(5)  5  2 or 7 f1(7)  7  2 or 5 coordinates of the vertices of a
Since f(x) and f1(x) are inverses, f(5)  7 and f1(7)  5. The inverse function can be
rectangle. Find the inverse of
found by exchanging the domain and range of the function. this relation and determine
whether the resulting ordered
Example 2 Find an Inverse Function pairs are also the coordinates
x6
of the vertices of a rectangle.
a. Find the inverse of f(x) 

. The inverse of the relation is
2
Step 1 Replace f(x) with y in the original equation. {(3, 1), (3, 6), (0, 6), (0, 1)}.
x6 x6 These ordered pairs also describe
f(x) 

y 


2 2 the vertices of a rectangle.
Step 2 Interchange x and y.
y6
x 

2
2
Step 3 Solve for y. a. Find the inverse of f(x) 
y6 1
x 

Inverse 
x  1. f
1(x) 
2x  2
2 2
2x  y  6 Multiply each side by 2.
b. Graph the function and its
2x  6  y Subtract 6 from each side.
inverse.
Step 4 Replace y with f1(x).
f (x )
y  2x  6 f 1(x)  2x  6
x6
The inverse of f(x) 

is f 1(x)  2x  6.
2
f (x) 
1–2x  1
b. Graph the function and its inverse. f (x )
7

1 O x
Graph both functions on the coordinate plane. 6 f (x )  2 x
6
5
The graph of f 1(x)  2x  6 is the reflection 4
x6
of the graph of f(x) 

over the line y  x. 3 f
1(x) 
2x  2
2 x 6
2 f (x ) 
1 2
O
1 2 3 4 5 6 7 x

INVERSES OF RELATIONS AND FUNCTIONS You can determine

whether two functions are inverses by finding both of their compositions. If both
equal the identity function I(x)  x, then the functions are inverse functions.

Inverse Functions
• Words Two functions f and g are inverse functions if and only if both of their
compositions are the identity function.
• Symbols [f  g](x)  x and [g  f](x)  x

www.algebra2.com/extra_examples Lesson 7-8 Inverse Functions and Relations 391

Differentiated Instruction
Logical After completing Example 3 on p. 392 and discussing how to
determine if two functions are inverses, challenge students to find two
functions, f and g, such that f(g(x))  g(f(x)), with one of the two
compositions having a value of x. You may wish to have students work
in groups to brainstorm as they attempt this puzzler.

Lesson 7-8 Inverse Functions and Relations 391

INVERSES OF RELATIONS Study Tip Example 3 Verify Two Functions are Inverses
AND FUNCTIONS Inverse Functions
1
Determine whether f(x)  5x  10 and g(x) 

x
2 are inverse functions.
5
Both compositions of f(x) Check to see if the compositions of f(x) and g(x) are identity functions.
In-Class Example Power
Point®
and g(x) must be the
identity function for f(x) [f  g](x)  f[g(x)] [g  f](x)  g[f(x)]
and g(x) to be inverses. It
3 Determine whether f(x)  is necessary to check them 
1
 f

x  2
5   g(5x  10)
3 4 both.

x  6 and g(x) 
x  8
4 3 1

 5

x  2  10
5
1


(5x  10)  2
5
are inverse functions. The  x  10  10 x22
functions are inverses since both x x
[f
g](x) and [g
f ](x) equal x. The functions are inverses since both [f  g](x) and [g  f](x) equal x.
Teaching Tip Have students
verify the results of Example 3 by
graphing the two functions on You can also determine whether two functions are inverse functions by graphing.
their graphing calculators and The graphs of a function and its inverse are mirror images with respect to the graph
of the identity function I(x)  x.
checking that the graphs are
reflections over the line y  x.

Inverses of Functions

3 Practice/Apply •
•
Use a full sheet of grid paper. Draw and label the x- and y-axes.
Graph y  2x  3.
• On the same coordinate plane, graph y  x as a dashed line.
• Place a geomirror so that the drawing edge is on the line y  x. Carefully
plot the points that are part of the reflection of the original line. Draw a line
Study Notebook through the points.
Have students— Analyze
• add the definitions/examples of x3
1. What is the equation of the drawn line? y 


2
the vocabulary terms to their 2. They are inverses. 2. What is the relationship between the line y  2x  3 and the line that you
f (x)  2x
3, drew? Justify your answer.
Vocabulary Builder worksheets for x3
g(x) 

3. Try this activity with the function y  x. Is the inverse also a function?
Chapter 7. 2 Explain. No; the graph does not pass the vertical line test.
• write the definitions of inverse
x3
f [g(x)]  2



2  
3x
relations and inverse functions.
g[f (x)] 
• include any other item(s) that they (2x
3)  3
When the inverse of a function is a function, then the original function is said to


 x be one-to-one . To determine if the inverse of a function is a function, you can use
find helpful in mastering the skills 2
the horizontal line test.
in this lesson.
f (x ) f (x )

O x O x
About the Exercises…
Organization by Objective
• Find Inverses: 14–31
• Inverses of Relations and
Functions: 32–37 No horizontal line can be drawn so that it A horizontal line can be drawn that passes
passes through more than one point. The through more than one point. The inverse
Odd/Even Assignments inverse of this function is a function. of this function is not a function.

Exercises 14–37 are structured 392 Chapter 7 Polynomial Functions

so that students practice the
same concepts whether they
are assigned odd or even
problems. Algebra Activity
Assignment Guide Materials: grid paper, geomirror
Basic: 15–37 odd, 41, 44–61 • Watch for students who mistakenly place the geomirror on the graph of the
function instead of the line y  x.
Average: 15–37 odd, 41–61
• Before students attempt Exercise 3, you may want to provide a brief review
Advanced: 14–36 even, 38–40, of the graphs of absolute value functions.
42–55 (optional: 56–61)

392 Chapter 7 Polynomial Functions

 NAME ______________________________________________ DATE ____________ PERIOD _____

Study
7-8 Guide
Study andIntervention
Guide and Intervention,
p. 417
Inverse(shown)
Functions andand p. 418
Relations
Concept Check 1. Determine whether f(x)  3x  6 and g(x)  x  2 are inverses. no Find Inverses
Two relations are inverse relations if and only if whenever one relation contains the

2. Switch x and y in 2. Explain the steps you would take to find an inverse function.
Inverse Relations

Property of Inverse
element (a, b), the other relation contains the element (b, a).
Suppose f and f 1 are inverse functions.
the equation and solve Functions Then f(a)  b if and only if f 1(b)  a.

for y. 3. OPEN ENDED Give an example of a function and its inverse. Verify that the Example Find the inverse of the function f(x)   x
 . Then graph the
2 1

3. Sample answer: two functions are inverses. function and its inverse.
5 5

f (x)  2x, f
1(x) 

Step 1 Replace f(x) with y in the original equation. f (x )
2 1 2 1
f(x) 

x 
→ y 

x 

0.5x; f [f
1(x)] 
5 5 5 5
4. Determine the values of n for which f(x)  xn has an inverse that is a function. Step 2 Interchange x and y.
f (x)  2–5x
1–5

f
1[f (x)]  x
O x
2 1
Assume that n is a whole number. n is an odd whole number. x 

y 

5 5
f –1(x)  5–2x  1–2

Lesson 7-8
Step 3 Solve for y.
2 1
x 

y 
Inverse
5 5
5x  2y  1
Guided Practice Find the inverse of each relation. 5x  1  2y
1
Multiply each side by 5.
Add 1 to each side.

(5x  1)  y Divide each side by 2.

GUIDED PRACTICE KEY 5. {(2, 4), (3, 1), (2, 8)} 6. {(1, 3), (1, 1), (1, 3), (1, 1)} 2
2
The inverse of f(x) 

x 
is f 1(x) 
(5x  1).
1 1

{(4, 2), (1,
3), (8, 2)}

5 5 2

Exercises Examples {(3, 1), (
1, 1), (
3, 1), (1, 1)}
Find the inverse of each function. Then graph the function and its inverse. Exercises
5, 6 1 1
f
1(x)
Find the inverse of each function. Then graph the function and its inverse.

7–9, 12, 2 7. f(x)  x 
x 8. g(x)  3x  1 1 19. y  2 x  5

1. f(x) 
x  1
2
2. f(x)  2x  3 3. f(x) 
x  2
1

7–9. See pp. 407A–407H for graphs. g
1(x) 

x

3 4

13 3 3 y  2x
10 f
1(x)   x  
3
2
3
2
f
1(x)   x  
1
2
3
2
f
1(x)  4x  8

10, 11 3 Determine whether each pair of functions are inverse functions. f (x )

f –1(x)  1–2x  3–2

f (x ) f (x )

f –1(x)  3–2x  3–2 f (x)  1–4x
2

10. f(x)  x  7 11. g(x)  3x  2 O x O x

O x

g(x)  x  7 yes x2 f (x)  2–3x
1

f(x) 
3
no f (x)  2x
3 f –1(x)  4x  8

Application PHYSICS For Exercises 12 and 13, use the following information.
The acceleration due to gravity is 9.8 meters per second squared (m/s2). To Gl NAME
/M G ______________________________________________
Hill 417 DATE ____________
GlPERIOD
Al _____
b 2

convert to feet per second squared, you can use the following chain of operations: Skills
7-8 Practice,
Practice (Average)
p. 419 and
9.8 m 100 cm 1 in. 1 ft Practice, p. 420 and (shown)



s2






.
1m 2.54 cm 12 in.
12. 32.2 ft/s2 Inverse Functions Relations
Find the inverse of each relation.

1. {(0, 3), (4, 2), (5, 6)} 2. {(5, 1), (5, 1), (5, 8)}
12. Find the value of the acceleration due to gravity in feet per second squared. {(3, 0), (2, 4), (
6, 5)} {(1,
5), (
1,
5), (8,
5)}
3. {(3, 7), (0, 1), (5, 9), (7, 13)} 4. {(8, 2), (10, 5), (12, 6), (14, 7)}
13. An object is accelerating at 50 feet per second squared. How fast is it {(
7,
3), (
1, 0), (9, 5), (13, 7)} {(
2, 8), (5, 10), (6, 12), (7, 14)}

accelerating in meters per second squared? 15.24 m/s2 5. {(5, 4), (1, 2), (3, 4), (7, 8)}
{(
4,
5), (2, 1), (4, 3), (8, 7)}
6. {(3, 9), (2, 4), (0, 0), (1, 1)}
{(9,
3), (4,
2), (0, 0), (1, 1)}

Find the inverse of each function. Then graph the function and its inverse.
3
7. f(x) 

x 8. g(x)  3  x 9. y  3x  2
4
4 x2
f
1(x)  x g
1(x)  x
3
Practice and Apply 3
f (x ) g (x )
y  
3
y

Homework Help Find the inverse of each relation. 14–19. See margin. O x O x O x

For See 14. {(2, 6), (4, 5), (3, 1)} 15. {(3, 8), (4, 2), (5, 3)}
Exercises Examples
14–19 1 16. {(7, 4), (3, 5), (1, 4), (7, 5)} 17. {(1, 2), (3, 2), (1, 4), (0, 6)} Determine whether each pair of functions are inverse functions.

10. f(x)  x  6 yes 11. f(x)  4x  1 yes 12. g(x)  13x  13 no
20–31, 2 1 1
19. {(2, 8), (6, 5), (8, 2), (5, 6)}
g(x)  x  6 g(x) 

(1  x) h(x) 

x  1
38–43 18. {(6, 11), (2, 7), (0, 3), (5, 3)} 4
6
13

13. f(x)  2x no 14. f(x) 

x yes 15. g(x)  2x  8 yes

7
32–37 3 g(x)  2x
7
g(x) 

x h(x) 

x  4
1
6 2
Find the inverse of each function. Then graph the function and its inverse. 16. MEASUREMENT The points (63, 121), (71, 180), (67, 140), (65, 108), and (72, 165) give
Extra Practice the weight in pounds as a function of height in inches for 5 students in a class. Give the

See page 844. 20. y  3 x 
3 21. g(x)  2x 22. f(x)  x  5 points for these students that represent height as a function of weight.
(121, 63), (180, 71), (140, 67), (108, 65), (165, 72)

20–28. See pp. REMODELING For Exercises 17 and 18, use the following information.

407A–407H for graphs. 23. g(x)  x  4 24. f(x)  3x  3 25. y  2x  1 The Clearys are replacing the flooring in their 15 foot by 18 foot kitchen. The new flooring
costs $17.99 per square yard. The formula f(x)  9x converts square yards to square feet.
x
17. Find the inverse f 1(x). What is the significance of f1(x) for the Clearys? f
1(x)  ;
1 1 5 1 9
21. g
1 ( x ) 


x 26. y 

x
3
y  3x 27. f(x) 

x
8
28. f(x) 

x  4
3
It will allow them to convert the square footage of their kitchen floor to
square yards, so they can then calculate the cost of the new flooring.
2
22. f
1(x)  x  5
18. What will the new flooring cost the Cleary’s? $539.70
4 2x  3 7x  4
29. f(x) 

x  7 30. g(x) 

31. f(x) 

NAME ______________________________________________
420 DATE ____________
Gl PERIOD
Al _____

23. g
1(x)  x
4
Gl /M G Hill b 2

5 6 8 Reading
7-8 Readingto
to Learn
Learn Mathematics
1 Mathematics, p. 421 ELL
24. f
1(x) 

x
1 Inverse Functions and Relations
3 Determine whether each pair of functions are inverse functions. Pre-Activity How are inverse functions related to measurement conversions?
1 1
25. y 


x



Read the introduction to Lesson 7-8 at the top of page 390 in your textbook.

2 2 32. f(x)  x  5 33. f(x)  3x  4 34. f(x)  6x  2 A function multiplies a number by 3 and then adds 5 to the result. What does
the inverse function do, and in what order? Sample answer: It first
8
f
1(x) 

x g(x)  x  5 yes g(x)  3x  4 no 1 subtracts 5 from the number and then divides the result by 3.
27. g(x)  x 

no
5 3 Reading the Lesson
28. f
1(x)  3x
12 35. g(x)  2x  8 36. h(x)  5x  7 37. g(x)  2x  1 1. Complete each statement.
range
x1
a. If two relations are inverses, the domain of one relation is the of
29–31. See pp. 1
f(x) 

x  4 yes
1
g(x) 

(x  7) yes f(x) 

yes the other.
b. Suppose that g(x) is a relation and that the point (4, 2) is on its graph. Then a point
407A–407H. 2 5 2 on the graph of g1(x) is (
2, 4) .
c. The horizontal line test can be used on the graph of a function to determine
www.algebra2.com/self_check_quiz Lesson 7-8 Inverse Functions and Relations 393

Lesson 7-8
whether the function has an inverse function.
d. If you are given the graph of a function, you can find the graph of its inverse by
reflecting the original graph over the line with equation yx .
e. If f and g are inverse functions, then (f  g)(x)  x and
(g  f)(x)  x .
NAME ______________________________________________ DATE ____________ PERIOD _____

Answers Enrichment,
7-8 Enrichment p. 422
f. A function has an inverse that is also a function only if the given function is
one-to-one .
g. Suppose that h(x) is a function whose inverse is also a function. If h(5)  12, then

14. {(6, 2), (5, 4), (
1,
3)} Miniature Golf

h1(12)  5 .

2. Assume that f(x) is a one-to-one function defined by an algebraic equation. Write the four
In miniature golf, the object of the game is to roll the golf ball into steps you would follow in order to find the equation for f 1(x).

15. {(8, 3), (
2, 4), (
3, 5)} the hole in as few shots as possible. As in the diagram at the right,
the hole is often placed so that a direct shot is impossible. Reflections
Ball
1. Replace f(x) with y in the original equation.
can be used to help determine the direction that the ball should be 2. Interchange x and y.

16. {(
4, 7), (5, 3), (4,
1), (5, 7)} rolled in order to score a hole-in-one.
3. Solve for y.
Hole
4. Replace y with f
1(x).

17. {(
2,
1), (
2,
3), (
4,
1), (6, 0)} Example 1 Helping You Remember
Using wall 
EF, find the path to use to Ball

18. {(11, 6), (7,
2), (3, 0), (3,
5)} score a hole-in-one.
Find the reflection image of the “hole” with respect to E F and label
3. A good way to remember something new is to relate it to something you already know.
How are the vertical and horizontal line tests related? Sample answer: The vertical
line test determines whether a relation is a function because the ordered
it H. The intersection of 
BH with wall 
EF is the point at which the

19. {(8, 2), (5,
6), (2, 8), (
6, 5)} shot should be directed. pairs in a function can have no repeated x-values. The horizontal line test
Hole determines whether a function is one-to-one because a one-to-one
function cannot have any repeated y-values.
E F

Lesson 7-8 Inverse Functions and Relations 393

 NUMBER GAMES For Exercises 38–40, use the following information.

4 Assess Damaso asked Sophia to choose a number between 1 and 20. He told her to add 7 to
that number, multiply by 4, subtract 6, and divide by 2.
4(x  7)
6
38. Write an equation that models this problem. y 


2
Open-Ended Assessment 1 11
39. Find the inverse. y 

x



2 2
Modeling On a large coordinate 40. Sophia’s final number was 35. What was her original number? 12
grid, have students model the
41. SALES Sales associates at Electronics Unlimited earn $8 an hour plus a 4%
graph of the identity function commission on the merchandise they sell. Write a function to describe their
f(x)  x using a length of string, a income, and find how much merchandise they must sell in order to earn $500
piece of raw spaghetti, or some- in a 40-hour week. I(m)  320  0.04m; $4500
thing similar. Then place a second TEMPERATURE For Exercises 42 and 43, use the following information.
length of string or spaghetti to Temperature 5
A formula for converting degrees Fahrenheit to Celsius is C(x) 

(x  32).
model the graph of a function. 9
The Fahrenheit
42. Find the inverse C1(x). Show that C(x) and C1(x) are inverses.
Have students model the graph temperature scale was
of the inverse of this function. established in 1724 by a 43. Explain what purpose C1(x) serves. 42–44. See margin.
physicist named Gabriel
Daniel Fahrenheit. The 44. CRITICAL THINKING Give an example of a function that is its own inverse.
Celsius temperature scale
Getting Ready for was established in the 45. WRITING IN MATH Answer the question that was posed at the beginning of
same year by an the lesson. See margin.
Lesson 7-9 astronomer named
Anders Celsius. How are inverse functions related to measurement conversions?
PREREQUISITE SKILL Students will
Source: www.infoplease.com Include the following items in your answer:
graph square root functions and
• an explanation of why you might want to know the customary units if you are
inequalities in Lesson 7-9. Stu- given metric units even if it is not necessary for you to perform additional
dents will need to solve radical calculations, and
equations. Use Exercises 56–61 to • a demonstration of how to convert the speed of light c  3.0  108 meters per
determine your students’ second to miles per hour.
familiarity with solving radical 3x  5
equations. Standardized 46. Which of the following is the inverse of the function f(x) 

? A
2
Test Practice A
2x  5
g(x) 

B g(x) 


3x  5 C g(x)  2x  5 D
2x  5
g(x) 


3 2 3

Answers 47. For which of the following functions is the inverse also a function? B
I. f(x)  x3 II. f(x)  x4 III. f(x)  x
9
42. C
1(x)   x  32; A I and II only B I only C I, II, and III D III only
5
C[C
1(x)]  C
1[C(x)]  x
43. It can be used to convert Celsius Maintain Your Skills
to Fahrenheit.
Mixed Review Find [g
h](x) and [h
g](x). (Lesson 7-7)
44. Sample answer: f(x)  x and
48. g[h(x)]  4x  48. g(x)  4x 49. g(x)  3x  2 50. g(x)  x  4
f
1(x)  x or f(x) 
x and 20, h[g(x)]  4x  5 h(x)  x  5 h(x)  2x  4 h(x)  x2  3x  28
f
1(x) 
x 49. g[h(x)]  6x
Find all of the rational zeros of each function. (Lesson 7-6)
45. Inverses are used to convert 10, h[g(x)]  6x
51. f(x)  x3  6x2  13x  42
7,
2, 3 52. h(x)  24x3  86x2  57x  20
between two units of 50. g[h(x)]  x2
3x 1 4 5

24, h[g(x)]  x2 


,

,


measurement. Answers should Evaluate each expression. (Lesson 5-7) 4 3 2
include the following. 5x
14 3



1



1



4



33
53. 16 2 64 54. 64 3 • 64 2 32 55. 1 3



• Even if it is not necessary, it is 8112
helpful to know the imperial Getting Ready for PREREQUISITE SKILL Solve each equation.
units when given the metric the Next Lesson (To review solving radical equations, see Lesson 5-8.)

units because most 25 56. x  5  3 4 57. 

x  4  11 117   2 196
58. 12  x
60.


8 25
measurements in the U.S. are 59. 
x  5  
2x  2
760. 
x  3  2  x 61. 3  x
  
x  6


4
given in imperial units so it is 394 Chapter 7 Polynomial Functions
easier to understand the
quantities using our system.
• To convert the speed of light from
meters per second to miles per hour,
3.0  108 meters 3600 seconds 1 mile
f(x)      
1 second 1 hour 1600 meters
 675,000,000 mi/hr

394 Chapter 7 Polynomial Functions

 Square Root Functions Lesson
and Inequalities Notes

• Graph and analyze square root functions.

• Graph square root inequalities.
1 Focus
Vocabulary are square root functions used in bridge design?
• square root function The Sunshine Skyway Bridge across Tampa
5-Minute Check
• square root inequality Bay, Florida, is supported by 21 steel cables, Transparency 7-9 Use as a
each 9 inches in diameter. The amount of quiz or review of Lesson 7-8.
weight that a steel cable can support is given
by w  8d2, where d is the diameter of the Mathematical Background notes
cable in inches and w is the weight in tons. If are available for this lesson on
you need to know what diameter a steel cable
should have to support a given weight, you p. 344D.
can use the equation d 

w8
. are square root
functions used in
bridge design?
SQUARE ROOT FUNCTIONS If a function contains a square root of a
variable, it is called a square root function . The inverse of a quadratic function Ask students:
is a square root function only if the range is restricted to nonnegative numbers.
• Using the given data and for-
y y mula, how much total weight
y  x2 y  x2 can be supported by the bridge
cables on the Sunshine Skyway
Bridge? 13,608 tons
O x O
y  x
x • If the bridge had been designed
to support 20,000 tons, how
y  x many 9-inch cables would have
been used in the bridge design?
y  x is not a function. y  x is a function. 31 cables
In order for a square root to be a real number, the radicand cannot be negative.
When graphing a square root function, determine when the radicand would be
negative and exclude those values from the domain.
2 Teach
Example 1 Graph a Square Root Function
Graph y 

3x  4. State the domain, range, x- and y-intercepts. SQUARE ROOT FUNCTIONS
Since the radicand cannot be negative, identify the domain.
3x  4 0 Write the expression inside the radicand as 0.
Teaching Tip In the graph on the
4 y
right, point out the two graphs and
x 

Solve for x. x y
lead students to see how the
3
4
4
The x-intercept is 

. 

0 domain and range of the graph of
3
3
Make a table of values and graph the 1 1 y  3 x  4 y  x have been restricted.
function. From the graph, you can see O x
0 2
4
that the domain is x 

, and the range 2 3.2
3
is y 0. The y-intercept is 2. 4 4

Lesson 7-9 Square Root Functions and Inequalities 395

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 7 Resource Masters 5-Minute Check Transparency 7-9
• Study Guide and Intervention, pp. 423–424 Answer Key Transparencies
• Skills Practice, p. 425
• Practice, p. 426 Technology
• Reading to Learn Mathematics, p. 427 Interactive Chalkboard
• Enrichment, p. 428
• Assessment, p. 444

Lesson x-x Lesson Title 395

In-Class Examples Power Example 2 Solve a Square Root Problem
Point®
SUBMARINES A lookout on a submarine is h feet above the surface of the

1 Graph y   3

x  1. State
2
water. The greatest distance d in miles that the lookout can see on a clear day
3
is given by the square root of the quantity h multiplied by

.
2
the domain, range, x- and a. Graph the function. State the domain and range.
y-intercepts.
The function is d 


3h

. Make a table of values and graph the function.
2
y
h d d

y 
3–
2x
1 0 0
2
3h
d
2 3 or 1.73
4 6 or 2.45
O x
6 9 or 3.00
Submarines 8  or 3.46
12 O h
Submarines were first used 10  or 3.87
15
by The United States in
2
The domain is x
 , and the 1776 during the The domain is h 0, and the range is d 0.
3 Revolutionary War.
range is y
0. The x-intercept Source: www.infoplease.com
b. A ship is 3 miles from a submarine. How high would the submarine have to
2 raise its periscope in order to see the ship?
is  . There is no y-intercept.
3
d


3h
2

Original equation
2 PHYSICS When an object is
spinning in a circular path of 3


3h
2

Replace d with 3.
radius 2 meters with velocity
3h
v, in meters per second, the 9 

Square each side.
2
centripetal acceleration a, in 18  3h Multiply each side by 2.
meters per second squared, is 6h Divide each side by 3.
directed toward the center of The periscope would have to be 6 feet above the water. Check this result on the
the circle. The velocity v and graph.
acceleration a of the object
are related by the function Graphs of square root functions can be transformed just like quadratic
v  2a. functions.
a. Graph the function. State the
domain and range. The domain
is a
0, and the range is v
0. Square Root Functions
1. D: x
0, R: y
0;
v
D: x
0, R: y
1; You can use a TI-83 Plus graphing calculator to graph square root functions.
D: x
0, R: y

2; Use 2nd [x ] to enter the functions in the Y list.
v 

2a Graphs are the same Think and Discuss 1–3. See pp. 407A–407H for graphs.
except they are trans- 1. Graph y  x, y  x + 1, and y  x  2 in the viewing window
lated vertically. [2, 8] by [4, 6]. State the domain and range of each function and
2. D: x
0, R: y
0; describe the similarities and differences among the graphs.
O a
D: x
0, R: y
0; 2. Graph y  x, y   2x, and y  8x in the viewing window [0, 10] by
D: x
0, R: y
0; [0, 10]. State the domain and range of each function and describe the
similarities and differences among the graphs.
Graphs are the same
b. What would be the centripetal except they get 3. Make a conjecture on how you could write an equation that translates the
increasingly less parent graph y  x to the left three units. Test your conjecture with the
acceleration of an object spin- graphing calculator. y 

ning along the circular path steep. x3
with a velocity of 4 meters 396 Chapter 7 Polynomial Functions
per second? 8 m/s2

Square Roots Students who have worked with non-graphing calculators will
likely be used to finding square roots by typing a value first and then pressing
the square root key. On a graphing calculator, the square root key is pressed
first, followed by the expression whose square root is to be found. Point out that
the calculator treats the radical symbol like an open set of parentheses. If students
want to graph y  
3


x  1, they need to enter a right parenthesis after the 1.
2

396 Chapter 7 Polynomial Functions

 SQUARE ROOT INEQUALITIES A square root inequality is an inequality SQUARE ROOT
involving square roots. You can use what you know about square root functions to INEQUALITIES
graph square root inequalities.
In-Class Example Power
Point®
Example 3 Graph a Square Root Inequality
a. Graph y 

2x
6.
3 a. Graph y  
3x  5.
y
Graph the related equation y  2x  6. Since y
the boundary should not be included, the graph
y  2 x
6
should be dashed.
The domain includes values for x 3, so the y 

3x  5
graph is to the right of x  3. Select a point and
test its ordered pair. O x
O x


Test (4, 1).
3
1  
2(4) 6 b. Graph y
4 
x.
2
1  2 true
y
Shade the region that includes the point (4, 1). y 
4  3–2x

b. Graph y


x  1.
Graph the related equation y  
x  1. y

The domain includes values for x 1, so the O x

graph includes x  1 and the values of x to the y  x  1
right of x  1. Select a point and test its ordered
pair.
Test (2, 1).
y 
x  1.
O x 3 Practice/Apply
1 
21
1 3 false
Study Notebook
Shade the region that does not include (2, 1).
Have students—
• complete the definitions/examples
for the remaining terms on their
Vocabulary Builder worksheets for
Chapter 7.
Concept Check 1. Explain why the inverse of y  3x2 is not a square root function. • include any other item(s) that they
1–2. See margin. 2. Describe the difference between the graphs of y  x  4 and y  
x  4.
find helpful in mastering the skills
in this lesson.
3. OPEN ENDED Write a square root function with a domain of {xx 2}.
Sample answer: y 

2x
4
Guided Practice Graph each function. State the domain and range of the function.
GUIDED PRACTICE KEY 4. y  x  2 D: x
0, R: y
2  D: x
0; R: y
0
5. y  4x
Exercises Examples 6. y  3  x D: x
0; R: y 3 7. y  
x  1  3 D: x
1; R: y
3
Answers
4–7 1 4–7. See pp. 407A–407H for graphs. 1. In order for it to be a square root
8–11 3 Graph each inequality. 8–11. See pp. 407A–407H.
12, 13 2
function, only the nonnegative
8. y

x41 9. y  
2x  4 range can be considered.
10. y  3  
5x  1 11. y 
x21 2. Both have the shape of the graph
www.algebra2.com/extra_examples Lesson 7-9 Square Root Functions and Inequalities 397 of y 
x, but y 
x
4 is
shifted down 4 units, and
y 
 x
4 is shifted to the right
Differentiated Instruction 4 units.

Auditory/Musical Divide the class into groups of 4 to 6 students.

Challenge each group to give themselves a rock or rap group name
based on the vocabulary in this lesson, such as “The Intercepts.” Have
students write a musical verse about some of the key facts in the
lesson, such as the domain, range, and intercepts of a graph.

Lesson 7-9 Square Root Functions and Inequalities 397

 Study Guide
NAME ______________________________________________ DATE

andIntervention
Intervention,
____________ PERIOD _____
Application FIREFIGHTING For Exercises 12 and 13, use the following information.
7-9 Study Guide and
p. 423 When fighting a fire, the velocity v of water being pumped into the air is the square
Square(shown) and
Root Functions p. 424
and Inequalities
Square Root Functions A function that contains the square root of a variable
root of twice the product of the maximum height h and g, the acceleration due to
expression is a square root function.
gravity (32 ft/s2).
Example Graph y 
 . State its domain and range.
3x
2
2
12. Determine an equation that will give the maximum height of the water as a
Since the radicand cannot be negative, 3x  2 0 or x

.
function of its velocity.
v 

3
2
The x-intercept is

. The range is y 0.
3
Make a table of values and graph the function. y
2gh
x y
13. The Coolville Fire Department must purchase a pump that is powerful enough
2



3
0
y 

3x
2 to propel water 80 feet into the air. Will a pump that is advertised to project
1
2
1
2
O x
water with a velocity of 75 ft/s meet the fire department’s need? Explain. Yes;
3 7

sample answer: the advertised pump will reach a maximum height of 87.9 ft.
Exercises
Graph each function. State the domain and range of the function.

1. y  2x
 2. y  3x
 3. y   
2x
Practice and Apply
y y y
O x

Lesson 7-9
y 

2x

y 
3

x
O x
Homework Help Graph each function. State the domain and range of each function.
–2x

14. y  3x 
15. y  5x 
16. y  4x
y

O x
For See
Exercises Examples
D: x
0; R: y
0 D: x
0; R: y 0 D: x
0; R: y 0 1
14–25 1 17. y 

x 18. y  
x2 19. y  
x7
4. y  2
x3 5. y  
2x  3 6. y  
2x  5 26–31 3 2
y y y
32–34 2 20. y  
2x  1 21. y  
5x  3 22. y  
x63
y  2

x
3

O x 23. y  5 
x4 24. y  
3x  6  4 25. y  2
3  4x  3
O x O
y 

2x  5
x Extra Practice
y 


2x
3
See page 844. 14–25. See pp. 407A–407H for graphs.
D: x
3; R: y
0
3
D: x
; R: y 0 D: x

; R: y
0
5
Graph each inequality. 26–31. See pp. 407A–407H.
14. D: x
0, R: y
0
2 2

Gl NAME
/M G ______________________________________________
Hill 423 DATE ____________
GlPERIOD
Al _____
b 2
15. D: x
0, R: y 0 26. y
6 x 27. y  
x5 28. y  
2x  8
Skills Practice, p. 425 and
7-9 Practice
Practice,
(Average)
p.Functions
426 (shown) 16. D: x
0, R: y 0 29. y 
5x  8 30. y 
x34 31. y  
6x  2  1
Square Root and Inequalities
Graph each function. State the domain and range of each function.
17. D: x
0, R: y
0
1. y  5x
 2. y  
x1 3. y  2
x2 18. D: x

2, R: y
0
y y y
19. D: x
7, R: y
0 32. ROLLER COASTERS The velocity of a roller coaster as it moves down a hill is
O x
20. D: x

0.5, R: y 0 v   , where v0 is the initial velocity and h is the vertical drop in
v02  64h
O x
21. D: x
0.6, R: y
0 feet. An engineer wants a new coaster to have a velocity of 90 feet per second
O x
22. D: x

6, R: y

3 when it reaches the bottom of the hill. If the initial velocity of the coaster at the
D: x
0, R: y
0 D: x
1, R: y 0 D: x

2, R: y
0
23. D: x

4, R: y 5 top of the hill is 10 feet per second, how high should the engineer make the hill?
4. y  
3x  4 5. y  
x74 6. y  1  
2x  3 24. D: x
2, R: y
4 125 ft
25. D: x 0.75, R: y
3
y y y

O x O x
AEROSPACE For Exercises 33 and 34, use the following information.
O x
The force due to gravity decreases with the square of the distance from the center
4
D: x
, R: y
0 D: x

7, R: y

4 D: x

, R: y 1
3 of Earth. So, as an object moves further from Earth, its weight decreases. The radius
3 2
of Earth is approximately 3960 miles. The formula relating weight and distance is

  3960, where W
Graph each inequality.
39602WE
7. y 6x
 8. y

y
x53 9. y  2
3x  2
y r

E represents the weight of a body on Earth, WS
y

x O x
WS
O

represents the weight of a body a certain distance from the center of Earth, and r
O x
represents the distance of an object above Earth’s surface.
10. ROLLER COASTERS The velocity of a roller coaster as it moves down a hill is
33. An astronaut weighs 140 pounds on Earth and 120 pounds in space. How far is
v   , where v0 is the initial velocity and h is the vertical drop in feet. If
v02  64h
v  70 feet per second and v0  8 feet per second, find h. about 75.6 ft
he above Earth’s surface? 317.29 mi
11. WEIGHT Use the formula d   39602 WE



Ws
 3960, which relates distance from Earth d 34. An astronaut weighs 125 pounds on Earth. What is her weight in space if she is
in miles to weight. If an astronaut’s weight on Earth WE is 148 pounds and in space Ws is
115 pounds, how far from Earth is the astronaut? about 532 mi Aerospace 99 miles above the surface of Earth? 119 lb
Gl NAME
/M G ______________________________________________
Hill 426 DATE ____________
Gl PERIOD
Al _____
b 2 The weight of a person is
Reading
7-9 Readingto
to Learn
Learn Mathematics equal to the product of the
Mathematics, p. 427 ELL 35. RESEARCH Use the Internet or another resource to find the weights, on Earth,
Square Root Functions person’s mass and the of several space shuttle astronauts and the average distance they were from
Pre-Activity How are square root functions used in bridge design? acceleration due to Earth’s
Read the introduction to Lesson 7-9 at the top of page 395 in your textbook. Earth during their missions. Use this information to calculate their weights
gravity. Thus, as a person
If the weight to be supported by a steel cable is doubled, should the
diameter of the support cable also be doubled? If not, by what number moves away from Earth,
while in orbit. See students’ work.
should the diameter be multiplied?
no;
2
 the person’s weight
decreases. However, mass 36. CRITICAL THINKING Recall how values of a, h, and k can affect the graph of a
Reading the Lesson remains constant. quadratic function of the form y  a(x  h)2  k. Describe how values of a, h,
1. Match each square root function from the list on the left with its domain and range from
the list on the right.

a. y  x
 iv i. domain: x 0; range: y 3
and k can affect the graph of a square root function of the form y  a 
x  h  k.
b. y  
x  3 viii ii. domain: x 0; range: y
0
See margin.
398 Chapter 7 Polynomial Functions
c. y  x
3 i iii. domain: x 0; range: y
3

d. y  
x3 v iv. domain: x 0; range: y 0

e. y  x
 ii v. domain: x 3; range: y 0

f. y  
x  3 vii vi. domain: x
3; range: y 3 NAME ______________________________________________ DATE ____________ PERIOD _____

g. y  
3  x  3 vi vii. domain: x 3; range: y
0
Enrichment,
7-9 Enrichment p. 428
h. y  x
  3 iii viii. domain: x 3; range: y 0

2. The graph of the inequality y


3x  6 is a shaded region. Which of the following Reading Algebra
points lie inside this region?
If two mathematical problems have basic structural similarities,
(3, 0) (2, 4) (5, 2) (4, 2) (6, 6)
they are said to be analogous. Using analogies is one way of
(3, 0), (5, 2), (4,
2) discovering and proving new theorems. c
p
The following numbered sentences discuss a three-dimensional b
analogy to the Pythagorean theorem. O q
Helping You Remember r
3. A good way to remember something is to explain it to someone else. Suppose you are 01 Consider a tetrahedron with three perpendicular faces that a
studying this lesson with a classmate who thinks that you cannot have square root meet at vertex O.
functions because every positive real number has two square roots. How would you 02 Suppose you want to know how the areas A, B, and C of the
explain the idea of square root functions to your classmate? three faces that meet at vertex O are related to the area D
of the face opposite vertex O.
Sample answer: To form a square root function, choose either the 03 It is natural to expect a formula analogous to the
positive or negative square root. For example, y 
x
 and y 

x are Pythagorean theorem z2  x2  y2, which is true for a
two separate functions. similar situation in two dimensions. z
04 To explore the three-dimensional case, you might guess a y
formula and then try to prove it.
05 Two reasonable guesses are D3  A3  B3  C3 and
D2  A2  B2  C2. O

398 Chapter 7 Polynomial Functions

 37. WRITING IN MATH Answer the question that was posed at the beginning of
the lesson. See margin.
About the Exercises…
How are square root functions used in bridge design?
Organization by Objective
Include the following in your answer:
• Square Root Functions:
• the weights for which a diameter less than 1 is reasonable, and
14–25
• the weight that the Sunshine Skyway Bridge can support.
• Square Root Inequalities:
Standardized 
38. What is the domain of f(x) 5x  3? C
26–31
Test Practice Odd/Even Assignments
A xx 
35
 B xx  
35
 C xx
35
 D xx 
35
 Exercises 14–31 are structured
39. Given the graph of the square root function at y so that students practice the
the right, which of the following must be true? D same concepts whether they
I. The domain is all real numbers. are assigned odd or even
II. The function is y  x  3.5. problems.
III. The range is {yy 3.5}. Alert! Exercise 35 involves
A I only B I, II, and III research on the Internet or
C II and III D III only O x other reference materials.
Assignment Guide
Basic: 15–31 odd, 33–49
Maintain Your Skills
Average: 15–31 odd, 33–49
Mixed Review Determine whether each pair of functions are inverse functions. (Lesson 7-8) Advanced: 14–32 even, 33–34,
3x  2
40. f(x)  3x yes 41. f(x)  4x  5 no 42. f(x) 

yes 36–49
7
1 1 5 7x  2
g(x) 

x g(x) 

x 

g(x) 
3

3 4 16

g
f
Find (f  g)(x), (f
g)(x), (f • g)(x), and

(x) for each f (x) and g(x). (Lesson 7-7)
43. f(x)  x  5
g(x)  x  3
44. f(x)  10x  20
g(x)  x  2
45. f(x)  4x2  9
g(x) 


1
4 Assess
43–45. See margin. 2x  3
Open-Ended Assessment
46. 4; If x is your 46. ENTERTAINMENT A magician asked a member of his audience to choose any Writing Have students write a
number, you can write number. He said, “Multiply your number by 3. Add the sum of your number
the expression and 8 to that result. Now divide by the sum of your number and 2.” The paragraph explaining why the
3x  x  8 magician announced the final answer without asking the original number. domain and range of square root


, which What was the final answer? How did he know what it was? (Lesson 5-4)
x2 functions and square root
equals 4 after dividing inequalities must be restricted.
the numerator and Simplify. (Lesson 5-2)
denominator by the 47. (x  2)(2x  8) 48. (3p  5)(2p  4) 49. (a2  a  1)(a  1)
GCF, x  2. 2x2
4x
16 6p2
2p
20 a3
1 Assessment Options
Quiz (Lesson 7-9) is available
on p. 444 of the Chapter 7
Resource Masters.

Population Explosion
It is time to complete your project. Use the information and data
you have gathered about the population to prepare a Web page.
Be sure to include graphs, tables, and equations in the
presentation.
www.algebra2.com/webquest Answers
x5
43. 2x  2; 8; x 2  2x
15;  ,
www.algebra2.self_check_quiz Lesson 7-9 Square Root Functions and Inequalities 399 x
3
x3
44. 11x
22; 9x
18;
Answers 10x 2
40x  40; 10, x  2
8x 3  12x 2
18x
26 3
36. If a is negative, the graph is reflected 37. Square root functions are used in bridge design 45.  , x 
 ;
2x  3 2
over the x-axis. The larger the value of because the engineers must determine what
a, the less steep the graph. If h is diameter of steel cable needs to be used to 8x 3  12x 2
18x
28 3
 , x 
 ;
2x  3 2
positive, the origin is translated to the support a bridge based on its weight. Answers
right, and if h is negative, the origin is should include the following. 3
2x
3, x 
 ;
translated to the left. When k is positive, 2
• Sample answer: When the weight to be
the origin is translated up, and when k is 3
supported is less than 8 tons. 8x 3  12x 2
18x
27, x 

2
negative, the origin is translated down. • 13,608 tons
Lesson 7-9 Square Root Functions and Inequalities 399
 Study Guide
and Review
Vocabulary and Concept Check
Vocabulary and
Complex Conjugates Theorem identity function (p. 391) quadratic form (p. 360)
Concept Check (p. 374) Integral Zero Theorem (p. 378) Rational Zero Theorem (p. 378)
composition of functions (p. 384) inverse function (p. 391) relative maximum (p. 354)
• This alphabetical list of degree of a polynomial (p. 346) inverse relation (p. 390) relative minimum (p. 354)
vocabulary terms in Chapter 7 depressed polynomial (p. 366) leading coefficients (p. 346) Remainder Theorem (p. 365)
includes a page reference Descartes’ Rule of Signs (p. 372) Location Principle (p. 353) square root function (p. 395)
where each term was end behavior (p. 348) one-to-one (p. 392) square root inequality (p. 397)
introduced. Factor Theorem (p. 366) polynomial function (p. 347) synthetic substitution (p. 365)
Fundamental Theorem of Algebra polynomial in one variable
• Assessment A vocabulary (p. 371) (p. 346)
test/review for Chapter 7 is
available on p. 442 of the Choose the letter that best matches each statement or phrase.
Chapter 7 Resource Masters. 1. A point on the graph of a polynomial function that has no other a. Complex Conjugates
nearby points with lesser y-coordinates is a ______. f Theorem
2. The ______ is the coefficient of the term in a polynomial function b. depressed polynomial
Lesson-by-Lesson with the highest degree. d c. inverse functions
Review 3. The ______ says that in any polynomial function, if an imaginary d. leading coefficient
number is a zero of that function, then its conjugate is also a zero. a e. quadratic form
For each lesson, 4. When a polynomial is divided by one of its binomial factors, the f. relative minimum
quotient is called a(n) ______. b
• the main ideas are 5. (x2)2  17(x2)  16  0 is written in ______. e
summarized, x2
6. f(x)  6x  2 and g(x) 

are ______ since [f o g](x) and [g o f](x)  x. c
6
• additional examples review
concepts, and
• practice exercises are provided.

Vocabulary 7-1 Polynomial Functions

PuzzleMaker See pages Concept Summary
346–352.
ELL The Vocabulary PuzzleMaker • The degree of a polynomial function in one variable is determined by the
greatest exponent of its variable.
software improves students’ mathematics
vocabulary using four puzzle formats— Example Find p(a  1) if p(x)  5x
x2  3x3.
crossword, scramble, word search using a p(a  1)  5(a  1)  (a  1)2  3(a  1)3 Replace x with a + 1.
word list, and word search using clues.  5a  5  (a2  2a  1)  3(a3  3a2  3a  1) Evaluate 5(a + 1), (a + 1)2,
Students can work on a computer screen and 3(a + 1)3.
or from a printed handout.  5a  5  a2  2a  1  3a3  9a2  9a  3
 3a3  8a2  12a  7 Simplify.
10. 21; x2  2xh  h2  5 11. 20; x2  2xh  h2
x
h 12.
129; 2x3  6x2h 
MindJogger Exercises Find p(
4) and p(x  h) for each function. 6xh2  2h3
1
Videoquizzes See Examples 2 and 3 on pages 347 and 348.
7. p(x)  x  2 8. p(x)  x  4 9. p(x)  6x  3
ELL MindJogger Videoquizzes
6; x 2 h
2 8;
x
2 h  4
21; 6x  6h  3
10. p(x)  x  5 11. p(x)  x  x 12. p(x)  2x3  1
provide an alternative review of concepts
presented in this chapter. Students work
in teams in a game show format to gain 400 Chapter 7 Polynomial Functions www.algebra2.com/vocabulary_review
points for correct answers. The questions
are presented in three rounds.
Round 1 Concepts (5 questions) TM

Round 2 Skills (4 questions) Have students look through the chapter to make sure they have
Round 3 Problem Solving (4 questions) included notes and examples for each lesson in this chapter in
their Foldable.
For more information
Encourage students to refer to their Foldables while completing
about Foldables, see
the Study Guide and Review and to use them in preparing for the
Teaching Mathematics
Chapter Test.
with Foldables.

400 Chapter 7 Polynomial Functions

 Chapter 7 Study Guide and Review Study Guide and Review

Answers
7-2 Graphing Polynomial Functions
See pages Concept Summary 14b. between
2 and
1, and
353–358.
• The Location Principle: Since zeros of a function are located at between
1 and 0
x-intercepts, there is also a zero between each pair of these zeros. 14c. Sample answer: no rel. max.,
• Turning points of a function are called relative maxima and relative minima. rel. min. at x 
1.2
Example Graph f(x)  x4
2x2  10x
2 by making a table of values. 15a. p (x )

Make a table of values for several x f(x ) f (x )

16 4
values of x and plot the points.
3 31
Connect the points with a smooth 8
2 14
8
4 O 4 8x
curve.
1 13 O x
4

4
2 2 4
0 2
8
8
f (x )  x 4
2 x 2  10 x
2
1 7

16 p (x )  x  x 4
2x 3  1
5

2 26
15b. between
2 and
3
Exercises For Exercises 13–18, complete each of the following. 15c. Sample answer: rel. max. at x  0
a. Graph each function by making a table of values. x 
1.6, rel. min. at x  0.8
b. Determine consecutive values of x between which each real zero is located. 16a. g (x )
c. Estimate the x-coordinates at which the relative maxima and relative minima
occur. See Example 1 on page 353. 13–18. See margin. 4

13. h(x)  x3  6x  9 14. f(x)  x4  7x  1

8
4 O 4 8x
15. p(x)  x  x  2x  1
5 4 3 16. g(x)  x3  x2  1

4
17. r(x)  4x  x  11x  3
3 2 18. f(x)  x3  4x2  x  2

8

g (x)  x 3
x 2  1

16b. between
1 and 0

7-3 Solving Equations Using Quadratic Techniques
See pages 16c. Sample answer: rel. max. at x 
Concept Summary
360–364. 0, rel. min. at x  0.7
• Solve polynomial equations by using quadratic techniques.
17a. r (x )

Example Solve x3
3x2
54x  0.

x3  3x2  54x  0 Original equation

x(x2  3x  54)  0 Factor out the GCF.

x
O
x(x  9)(x  6)  0 Factor the trinomial.

x0 or x90 or x60 Zero Product Property

5 r (x )  4x 3  x 2
11x  3
19.

,
3, 0 x  0 x9 x  6
3
20.
8, 0, 5 17b. between
2 and
1, between 0
Exercises Solve each equation. See Example 2 on page 361. and 1, and between 1 and 2
21. 4,
2 

2i
3 19. 3x3  4x2  15x  0 20. m4  3m3  40m2 21. a3  64  0 17c. Sample answer: rel. max. at
23. 2,
2 22. r  9r  8  23. x4  8x2  16  0

2

1

24. x 3  9x 3  20  0 64, 125 x 
1, rel. min. at x  0.9
18a. f (x )
Chapter 7 Study Guide and Review 401

O x
13a. h (x ) 13b. at x  3 14a. f (x )
12
x

8
4 O 4 8 13c. Sample answer: rel. max. at
8

4 x 
1.4, rel. min. at x  1.4
4 f (x )  x 3  4x 2  x
2

8
x

12
8
4 O 4 8 18b. between
4 and
3, at x 
1,
and between 0 and 1
h (x)  x 3
6x
9 f (x )  x 4  7x  1 18c. Sample answer: rel. max. at
x 
2.5, rel. min. at x 
0.1

Chapter 7 Study Guide and Review 401

Study Guide and Review Chapter 7 Study Guide and Review

7-4 The Remainder and Factor Theorems

See pages Concept Summary
365–370.
• Remainder Theorem: If a polynomial f(x) is divided by x  a, the
remainder is the constant f(a) and f(x)  q(x) (x  a)  f(a) where q(x) is a
polynomial with degree one less than the degree of f(x).
• Factor Theorem: x  a is a factor of polynomial f(x) if and only if f(a)  0.
Example Show that x  2 is a factor of x3
2x2
5x  6. Then find any remaining factors
of the polynomial.
2 1 2 5 6 The remainder is 0, so x  2 is a factor of x3  2x2  5x 
2 8 6 6. Since x3  2x2  5x  6  (x  2)(x2  4x  3), the
1 4 3 0 factors are of x3  2x2  5x  6 are (x  2)(x  3)(x  1).

Exercises Use synthetic substitution to find f(3) and f(
2) for each function.
See Example 2 on page 367. 26. 1, 16 27. 20,
20
25. f(x)  x2  5 4,
1 26. f(x)  x2  4x  4 27. f(x)  x3  3x2  4x  8

Given a polynomial and one of its factors, find the remaining factors of the
polynomial. Some factors may not be binomials. See Example 3 on page 367.
28. x3  5x2  8x  4; x  1 29. x3  4x2  7x  6; x  2 x2  2x  3

28. x  2, x  2

7-5 Roots and Zeros

See pages Concept Summary
371–377.
• Fundamental Theorem of Algebra: Every polynomial equation with
degree greater than zero has at least one root in the set of complex numbers.

• Use Descartes’ Rule of Signs to determine types of zeros of polynomial functions.

• Complex Conjugates Theorem: If a  bi is a zero of a polynomial function,
then a  bi is also a zero of the function.
Example State the possible number of positive real zeros, negative real zeros, and
imaginary zeros of f(x)  5x4  6x3
8x  12.
Since f(x) has two sign changes, there are 2 or 0 real positive zeros.
f(x)  5x4  6x3  8x  12 Two sign changes → 0 or 2 negative real zeros
There are 0, 2, or 4 imaginary zeros. 30. 3 or 1; 1; 2 or 0
33. 3 or 1; 1; 0 or 2 34. 2 or 0; 2 or 0; 4, 2, or 0 35. 2 or 0; 2 or 0; 4, 2, or 0
Exercises State the possible number of positive real zeros, negative real zeros,
and imaginary zeros of each function. See Example 2 on page 373.
30. f(x)  2x4  x3  5x2  3x  9 31. f(x)  7x3  5x  1 1; 0; 2
32. f(x)  4x4  x2  x  1 1; 1; 2 33. f(x)  3x4  x3  8x2  x  7
34. f(x)  x4  x3  7x  1 35. f(x)  2x4  3x3  2x2  3

402 Chapter 7 Polynomial Functions

402 Chapter 7 Polynomial Functions

 Chapter 7 Study Guide and Review Study Guide and Review

7-6 Rational Zero Theorem

See pages Concept Summary
378–382.
• Use the Rational Zero Theorem to find possible zeros of a polynomial function.
• Integral Zero Theorem: If the coefficients of a polynomial function are integers
such that a0  1 and an  0, any rational zeros of the function must be factors of an.
Examples Find all of the zeros of f(x)  x3  7x2
36.
There are exactly three complex zeros.
There are either one or three positive real zeros and two or zero negative real zeros.
The possible rational zeros are 1, 2, 3, 4, 6, 9, 12, 18, 36.
2 1 7 0 36 x3  7x2  36  (x  2)(x2  9x  18)
2 18 36  (x  2)(x  3)(x  6)
1 9 18 0 Therefore, the zeros are 2, 3, and 6.

Exercises Find all of the rational zeros of each function. See Example 3 on page 379.
1
36. f(x)  2x3  13x2  17x  12

2
, 3, 4 37. f(x)  x4  5x3  15x2  19x  8
38. f(x)  x3  3x2  10x  24
3, 2, 4 39. f(x)  x4  4x3  7x2  34x  24
40. f(x)  2x  5x  28x  15
3 2 41. f(x)  2x4  9x3  2x2  21x  10
1 1
37.
1,
1 39. 1, 2, 4,
3 40.
3, 5,

41.

, 2
2 2
7-7 Operations of Functions
See pages Concept Summary
383–389.

Operation Definition Operation Definition

Sum (f  g)(x)  f(x)  g(x) Quotient  
f f(x)


(x) 

, g(x)  0
g g(x)

Difference (f – g)(x)  f(x) – g(x) Composition [f  g](x)  f[g(x)]

Product (f g)(x)  f(x) g(x) — —

Example If f(x)  x2
2 and g(x)  8x
1. Find g[f(x)] and f[g(x)].

g[f(x)] 8(x2  2)  1 Replace f(x) with x2  2.

 8x2  16  1 Multiply.

43. x 2
1;  8x2  17 Simplify.

x 2
6x  11 f [g(x)] (8x  1)2 2 Replace g(x) with 8x  1.
44.
2x 2

 64x2  16x  1  2 Expand the binomial.
1; 4x 2
4x
2  64x2  16x  1 Simplify.
45.
15x

5;
15x  Exercises Find [g  h](x) and [h  g](x). See Example 4 on page 385.
25 42. h(x)  2x  1 6x  1; 43. h(x)  x2  2 44. h(x)  x2  1
46. x 3
2; g(x)  3x  4 6x  7 g(x)  x  3 g(x)  2x  1
x 3
6x 2  45. h(x)  5x 46. h(x)  x3 47. h(x)  x  4 x  4;
12x
8 g(x)  3x  5 g(x)  x  2 g(x)  x x  4

Chapter 7 Study Guide and Review 403

Chapter 7 Study Guide and Review 403

 • Extra Practice, see pages 842–844.
Study Guide and Review • Mixed Problem Solving, see page 868.

Answers 7-8 Inverse Functions and Relations

See pages Concept Summary
54. D: x

2, R: y
0 390–394.
• Reverse the coordinates of ordered pairs to find the inverse of a relation.
y
8 • Two functions are inverses if and only if both of their compositions
7 are the identity function. [f  g](x)  x and [g  f ](x)  x
6 • A function is one-to-one when the inverse of the function is a function.
5
4 Example Find the inverse of f(x) 
3x  1.
y  13 x  2
3 Rewrite f(x) as y  3x  1. Then interchange the variables and solve for y.
2 x  3y  1 Interchange the variables.
1
3y  x  1 Solve for y.

2 O 1 2 3 4 5 6x x  1
y 

Divide each side by 3.
3
1 x  1
3 f (x) 

Rewrite in function notation.
55. D: x
 , R: y
0 3
5
y Exercises Find the inverse of each function. Then graph the function and its
8 inverse. See Example 2 on page 391. 48–53. See pp. 407A–407H.
7 1
48. f(x)  3x  4 49. f(x)  2x  3 50. g(x) 

x  2
6 y  5x
3 3
3x  1
5 51. f(x) 

52. y  x2 53. y  (2x  3)2
2
4
3
2
1
x 7-9 Square Root Functions and Inequalities
O 1 2 3 4 5 6 7 8 See pages Concept Summary
395–399.
56. D: x
3, R: y
4 • Graph square root inequalities in a similar manner as graphing square
y root equations.
8
7 Example Graph y  2 

x
1.
6 y
5 x y 7
6 y  2  x
1
4 1 2
5
3 y  4  2x
3 2 3 4
2 3
3  or 3.4
2  2 2
1
x 4  or 3.7
2  3 1
O 1 2 3 4 5 6 7 8 O 1 2 3 4 5 6 7x
5 4
57. y
6
5 Exercises Graph each function. State the domain and range of each
4 function. See Examples 1 and 2 on pages 395 and 396. 54–56. See margin.
3 1
2 y  x
2 54. y 


x2 55. y  
5x  3 56. y  4  2
x3
3
1
Graph each inequality. See Example 3 on page 397. 57
58. See margin.
O 1 2 3 4 5 6 7 8x 57. y x  2 58. y  
4x  5

2
404 Chapter 7 Polynomial Functions

58. y
8
7
Answers (p. 405)
6 y  4x
5
5 20. x 2  2x
1
4
3 21.
x 2  2x
7
2 22. 2x 3
4x 2  6x
12
1
2x
4
23. 
O 1 2 3 4 5 6 7 8x x 3
2

24a. A  1000(1  r)6  1000(1  r)5  1000(1  r)4  1200(1  r)3  1200(1  r)2 
2000(1  r)

404 Chapter 7 Polynomial Functions

 Practice Test

Vocabulary and Concepts

Assessment Options
Match each statement with the term that it best describes.
Vocabulary Test A vocabulary
1. [f  g](x)  f [g(x)] b a. quadratic form
test/review for Chapter 7 can be
b. composition of functions
2. [f  g](x)  x and [g  f ](x) = x c found on p. 442 of the Chapter 7
c. inverse functions
3. x
2  2x  4  0 a Resource Masters.
Chapter Tests There are six
Skills and Applications Chapter 7 Tests and an Open-
Ended Assessment task available
For Exercises 4–7, complete each of the following. 4–7. See pp. 407A–407H.
in the Chapter 7 Resource Masters.
a. Graph each function by making a table of values.
b. Determine consecutive values of x between which each real zero is located.
c. Estimate the x-coordinates at which the relative maxima and relative minima occur. Chapter 7 Tests
4. g(x)   x3  6x  4
6x2 5. h(x)  
x4 
6x3 x
8x2 Form Type Level Pages
6. f(x)  x3  3x2  2x  1 7. g(x)  x4  2x3  6x2  8x  5 1 MC basic 429–430

Solve each equation. 8. 0,
4 
34 10. 
6, 
3 2A MC average 431–432
1
3

8. p3  8p2  18p 9. 16x4  x2  0 0, 

10. r 4  9r2  18  0 11. p 2  8  0 4 2B MC average 433–434
4
Given a polynomial and one of its factors, find the remaining factors of 2C FR average 435–436
the polynomial. Some factors may not be binomials. 2D FR average 437–438
12. x3  x2  5x  3; x  1 x
3, x  1 13. x3  8x  24; x  2 x 2
2x  12 3 FR advanced 439–440
State the possible number of positive real zeros, negative real zeros, and MC = multiple-choice questions
imaginary zeros for each function.
FR = free-response questions
14. f(x)  x3  x2  14x  24 2 or 0; 1; 0 or 2 15. f(x)  2x3  x2  16x  5 3 or 1; 0; 0 or 2

Find all of the rational zeros of each function.

Open-Ended Assessment
Performance tasks for Chapter 7
16. g(x)  x3  3x2  53x  9 9 17. h(x)  x4  2x3  23x2  2x  24
6, 4
can be found on p. 441 of the
Determine whether each pair of functions are inverse functions. Chapter 7 Resource Masters. A
x9 1 1 sample scoring rubric for these
18. f(x)  4x  9, g(x) 

no 19. f(x) 

, g(x) 

 2 yes
4 x2 x
tasks appears on p. A34.
If f(x)  2x
4 and g(x)  x2  3, find each value. 20
23. See margin.
g
f Unit 2 Test A unit test/review
20. ( f  g)(x) 21. ( f  g)(x) 22. ( f g)(x) 23.

(x)
can be found on pp. 449–450 of
24. FINANCIAL PLANNING Toshi will start college in six years. According the Chapter 7 Resource Masters.
to their plan, Toshi’s parents will save $1000 each year for the next three
years. During the fourth and fifth years, they will save $1200 each year. First Semester Test A test for
During the last year before he starts college, they will save $2000.
Chapters 1–7 can be found on
a. In the formula A  P(1  r)t, A  the balance, P  the amount invested,
r  the interest rate, and t  the number of years the money has been invested. pp. 451–454 of the Chapter 7
Use this formula to write a polynomial equation to describe the balance of the Resource Masters.
account when Toshi starts college. See margin.
b. Find the balance of the account if the interest rate is 6%. $8916.76
TestCheck and
? D
25. STANDARDIZED TEST PRACTICE Which value is included in the graph of y  2x
A (2, 2) B (–1, 1) C (0, 0) D None of these
Worksheet Builder
This networkable software has
www.algebra2.com/chapter_test Chapter 7 Practice Test 405
three modules for assessment.
• Worksheet Builder to make
worksheets and tests.
Portfolio Suggestion • Student Module to take tests
Introduction In mathematics, polynomial equations can be used to model on-screen.
many real-world problems. The solution to the polynomial equation provides a • Management System to keep
solution to the real-world problem. student records.
Ask Students From your work in this chapter, select a real-world problem
modeled by a polynomial equation and show how you solved it. Explain how
the solution to the polynomial equation relates to the solution of the real-world
problem. Place your work in your portfolio.

Chapter 7 Practice Test 405

 Standardized
Test Practice

These two pages contain practice 6. What is the midpoint of the line segment
Part 1 Multiple Choice whose endpoints are represented on the
questions in the various formats coordinate grid by the points (5, 3)
that can be found on the most Record your answers on the answer sheet and (1, 4)? B
provided by your teacher or on a sheet of
frequently given standardized paper.
tests. A 3, 
12
 B 3,
12

2 4 2
1. If



2  

3 , then what is the value of p? B
A practice answer sheet for these
p p p
1 1
C 2, 
72
 D 2,
12

two pages can be found on p. A1 A 1 B 1 C 

D


2 2
of the Chapter 7 Resource Masters. 7. For all n  0, what is the slope of the line
passing through (n, k) and (n, k)? D
NAME

Standardized
DATE PERIOD
2. There are n gallons of liquid available to fill a
7 Standardized Test Practice
Test Practice n k
Student Recording
Student Record Sheet,
Sheet (Use with pages 406–407 of p. A1Edition.)
the Student
tank. After k gallons of the liquid have filled A 0 B 1 C

D


the tank, how do you represent in terms of n k n
Part 1 Multiple Choice
Select the best answer from the choices given and fill in the corresponding oval.
and k the percent of liquid that has filled the
1 A B C D 4 A B C D 7 A B C D 10 A B C D tank? A 8. Which of the following is a quadratic
2 A B C D 5 A B C D 8 A B C D 11 A B C D

100k n equation in one variable? B

3 A B C D 6 A B C D 9 A B C D 12 A B C D A

% B

%
n 100k
Part 2 Short Response/Grid In
A 3(x  4)  1  4x  9
100n n


%

% 3x(x  4)  1  4x  9
Solve the problem and write your answer in the blank.
C D B
Also enter your answer by writing each number or symbol in a box. Then fill in
the corresponding oval for that number or symbol.
k 100(n  k)
13 15 17 19 C 3x(x2  4)  1  4x  9
.
/
.
0
/
.
0
.
0
.
/
.
0
/
.
0
.
0
.
/
.
0
/
.
0
.
0
.
/
.
0
/
.
0
.
0 3. How many different triangles have sides of D y  3x2  8x  10
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
2
3
4
lengths 4, 9 and s, where s is an integer and
4  s  9? D
5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

9. Simplify 
t3 
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
4 8
t 2. D
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

14 16 18 A 0 B 1 C 2 D 3

3

1

3

.
/
.
/
. . .
/
.
/
. . .
/
.
/
. .
A t 16 B t2 C t4 D t
0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2 2 2
3 3 3 3 3 3 3 3 3 3 3 3
4
5
6
4
5
6
4
5
6
4
5
6
4
5
4
5
4
5
4
5
4
5
4
5
4
5
4
5 4. Triangles ABC and DEF are similar. The area
of ABC is 72 square units. What is the
6 6 6 6 6 6 6 6

10. Which of the following is a quadratic

7 7 7 7 7 7 7 7 7 7 7 7
8 8 8 8 8 8 8 8 8 8 8 8
9 9 9 9 9 9 9 9 9 9 9 9

Part 3 Quantitative Comparison

perimeter of DEF? B 1 2
equation that has roots of 2

and

? C
2 3
Answers

Select the best answer from the choices given and fill in the corresponding oval. D
20 A B C D 22 A B C D 24 A B C D
A 5x2  11x  7  0
21 A B C D 23 A B C D

A B 5x2  11x  10  0
12 28
C 6x2  19x  10  0

Additional Practice D 6x2  11x  10  0

C B
See pp. 447–448 in the Chapter 7 F E
Resource Masters for additional 11. If f(x)  3x  5 and g(x)  2  x2, then what
standardized test practice. A 56 units B 28  282 units is equal to f [g(2)]? D
C 562 units D 28  142 units A 3 B 6 C 12 D 13

5. If 2  3x  1 and x  5  0, then x could 12. Which of the following is a root of

equal each of the following except A f(x)  x3  7x  6? B
A 5. B 4. C 2. D 0. A 1 B 2 C 3 D 6
406 Chapter 7 Polynomial Functions

TestCheck and
Log On for Test Practice
Worksheet Builder
The Princeton Review offers
additional test-taking tips and Special banks of standardized test
practice problems at their web site. Visit questions similar to those on the SAT,
www.princetonreview.com or ACT, TIMSS 8, NAEP 8, and Algebra 1
www.review.com End-of-Course tests can be found on
this CD-ROM.

406 Chapter 7 Polynomial Functions

 Aligned and
verified by

18. If the measures of the sides of a triangle are 3,

Part 2 Short Response/Grid In 8, and x and x is an integer, then what is the
least possible perimeter of the triangle? 17
Record your answers on the answer sheet
provided by your teacher or on a sheet of 19. If the operation ❖ is defined by the equation
paper. x ❖ y  3x  y, what is the value of w in the
equation w ❖ 6  2 ❖ w? 3
13. A group of 34 people is to be divided into
committees so that each person serves on
exactly one committee. Each committee Part 3 Quantitative Comparison
must have at least 3 members and not more
than 5 members. If N represents the Compare the quantity in Column A and the
maximum number of committees that can quantity in Column B. Then determine
be formed and n represents the minimum whether:
number of committees that can be formed,
what is the value of N  n? 4 A the quantity in Column A is greater,
14. Raisins selling for $2.00 per pound are to be B the quantity in Column B is greater,
mixed with peanuts selling for $3.00 per C the two quantities are equal, or
pound. How many pounds of peanuts are
needed to produce a 20-pound mixture that D the relationship cannot be determined
sells for $2.75 per pound? 15 from the information given.
15. The mean of 15 scores is 82. If the mean of 7
Column A Column B
of these scores is 78, what is the mean of the
remaining 8 scores? 85.5
20. D 2.5% of 10x 0.025x
16. Jars X, Y, and Z each contain 10 marbles.
What is the minimum number of marbles
that must be transferred among the jars so 21.
b˚
that the ratio of the number of marbles in jar B a˚ c˚ ᐉ
X to the number of marbles in jar Y to the
160˚ 120˚
number of marbles in jar Z is 1 : 2 : 3? 5 m
17. If the area of BCD is 40% of the area of
ABC, what is the measure of  ? 6
AD
A
ac b

22. B x0
3
D B x


3x
0.4
5

C
23. C y˚ t
s z˚

x˚ w
Test-Taking Tip yxz
Questions 13, 16, and 18 Words such as
maximum, minimum, least, and greatest indicate w 
s 2  t 2
that a problem may involve an inequality. Take
special care when simplifying inequalities that
involve negative numbers. 24. C 28 27  27

www.algebra2.com/standardized_test Chapters 7 Standardized Test Practice 407

Chapter 7 Standardized Test Practice 407

 Pages 350–352, Lesson 7-1 5. x f(x) f (x)
8
68. y 69. y 3 20
4
2 9
1 2
4
2 O 2 4x
yx 4
2
0 5
O x
4
1 0

8
O x 2 5
3 26 f (x )  x
7x 2  x  5
4

y 
x 2  6x
5
70. y 6. f (x) 7. f (x)
8

8
4 O 4 8x O x O x

4
f (x )  x 3
x 2  1
f (x )  x 4
4x 2  2
1
y  2 x 2  2x
6
8. f (x) 9. f (x)
8
8
Additional Answers for Chapter 7

Page 356, Lesson 7-2

4
Graphing Calculator Investigation 4
1. rel. max. at x  0,
4
2 O 2 4x
rel. min. at x  2
4
2 O 2 4x

4

4

8

f (x )  x 3  2x 2
3x
5 f (x )  x 4
8x 2  10

2. rel. max. at (0, 4) rel. min. at (2, 0) 10. c (x )

12000
3. Sample answer: two rel.
min. points at (0, 8) and 10000
(4.4, 25.8) and one rel.
Cable TV Systems

max. point at (1.6, 3.2) 8000

6000

4000 c (x) 
43.2t 2  1343t  790

Pages 356–358, Lesson 7-2
4. x f(x) f (x ) 2000
8
3 20
4 O t
2 0 4 8 12 16
1 6 Years Since 1985

4
2 O 2 4x
0 4 11. rel. max. between x  15 and x  16, and no rel. min.;

4
1 0 f(x) →  as x → , f(x) →  as x → .
f(x)  x 3
x 2
4x  4
2 0 12. The number of cable TV systems rose steadily from
1985 to 2000. Then the number began to decline.
3 10

407A Chapter 7 Additional Answers

13a. x f(x) f (x ) 17a. x f(x) f (x )
5 25 4 1 75
4
4 0 O 0 16

2 2 4x
3 9 1 3
4
2 O 2 4x

4
2 8 2 0
4
1 3
8 3 7

8
0 0 4 0
3 2
f (x) 
x
4x
1 5 5 39 f (x) 
3x 3  20x 2
36x  16
2 24
17b. between 0 and 1, at x  2, and at x  4
13b. at x  4 and x  0 17c. Sample answer: rel. max. at x  3, rel. min. at x  1
13c. Sample answer: rel. max. at x  0, rel. min. at x  3 18a. x f(x) f (x )
14a. x f(x) f (x ) 2 29 O x
2 10 8 1 8
1 3 4 0 1
0 6 1 2
1 5
4
2 O 2 4x 2 5
2 6
4 3 4

Additional Answers for Chapter 7

3 15 4 7 f (x )  x 3
4x 2 2x
1
f (x)  x 3
2x 2 6
4 38 5 34
14b. between 2 and 1 18b. between 3 and 4
3
14c. Sample answer: rel. max. at x  0, rel. min. at x 
18c. Sample answer: rel. max. at x  0.5, rel. min. at x  2.5
2
19a. x f(x) f (x )
15a. x f(x) f (x )
3 73 4
2 18 2 8
1 2
4
2 O 2 x
1 7
0 2
4
O x 0 8
1 0 1 7 f (x)  x 4
8
8
2 2 2 8
3 2 3 73
4 18 f (x)  x 3
3x 2 2
19b. between 2 and 1, and between 1 and 2
15b. at x  1, between 1 and 0, and between 2 and 3 19c. Sample answer: no rel. max., rel. min. at x  0
15c. Sample answer: rel. max. at x  0, rel. min. at x  2 20a. x f(x) f (x )
16
16a. x f(x) f (x )
3 0
5 9 2 15
8
4
4 7 1 0
4
2 O 2 x
3 9
8
4 O 4 8x 0 9
8
2 3
4 1 0
1 5
16
2 15

8
0 9 3 0 f (x )  x
10x 2 9
4

1 3 f (x )  x 3  5x 2
9
4 105
2 19
20b. at x  3, x  1, x  1, and x  3
16b. between 5 and 4, between 2 and 1, and 20c. Sample answer: rel. max. at x  0, rel. min. at
between 1 and 2 x  2 and x  2
16c. Sample answer: rel. max. at x  3, rel. min. at x  0

Chapter 7 Additional Answers 407B

 21a. x f(x) f (x ) 24c. Sample answer: rel. max. at x  0.5, rel. min. at
8
x  0.5 and at x  1.5
4 169
4 25a. x f(x) f (x )
3 31
2 7 4 77 24

4
2 O 2 4x
1 5 3 30 16

4
0 1 2 7

8 8
1 1 1 2
2 1 f (x ) 
x 4  5x 2
2x
1 0 3
4
2 O 2 4x
3 43 1 2
2 55 f (x)  x 5  4x 4
x 3
9x 2  3
21b. between 3 and 2, between 1 and 0, between 0
and 1, and between 1 and 2 25b. between 4 and 3, between 2 and 1, between
21c. Sample answer: rel. max. at x  2 and at x  1.5, 1 and 0, between 0 and 1, and between 1 and 2
rel. min. at x  0 25c. Sample answer: rel. max. at x  3 and at x  0, rel.
22a. x min. at x  1 and at x  1
f(x) f (x )
24 26a. x f(x) f (x )
3 39 40
2 5
16 2 88
20
1 3 8 1 5
0 3 0 6
4
2 4x
Additional Answers for Chapter 7

O 2

4
2 O 2 4x 1 5
1 5
20

8 2 20
2 21

40
3 15 f (x) 
x 4  x 3 8x 2
3 3 3
10 f (x)  x
6x  4x 3  17x 2
5x
6
5 4
4 67 4
5 269
22b. between 3 and 2, between 1 and 0, between 0
and 1, and between 3 and 4 26b. between 2 and 1, between 1 and 0, between 0
22c. Sample answer: rel. max. at x  1.5 and at x  2.5, and 1, between 2 and 3, and between 4 and 5
rel. min. at x  0 26c. Sample answer: rel. max. at x  1 and at x  2,
23a. x f (x )
rel. min. at x  0 and at x  3.5
f(x)
31. x 0 2 4 6 8 10 12 14 16 18 20
1 65 4

0 6 B(x) 25 34 40 45 50 54 59 64 68 71 71

2 O 2 4 x
G(x) 26 33 39 44 49 53 56 59 61 61 60
1 1

4
2 2 y B (x )
3 3
8 70
65
4 10
60
Average Height (in.)

f (x)  x 4
9x 3  25x 2
24x  6

5 11 55 G (x )
50
23b. between 0 and 1, between 1 and 2, between 2 and 3,
45
and between 4 and 5
40
23c. Sample answer: rel. max. at x  2, rel. min. at 35
x  0.5 and at x  4 30
24a. x f(x) f (x ) 25
4
20
2 45
1 4
2 O 2 4 6x 0 2 4 6 8 10 12 14 16 18 x
Age (yrs)
0 5
4

1 6
8
2 7
f (x)  2x 4
4x 3
2x 2  3x
5
3 40

24b. between 2 and 1, and between 2 and 3

407C Chapter 7 Additional Answers

36. y 37. y

O x O x

[1930, 2010] scl: 10 by [0, 200] scl: 20

38. y 39. y Pages 362–364, Lesson 7-3

17. 0, 4, 3
18. 0, 1, 5
O x O x 19.  3, 
3, i  3, i 
3
20. 0, 4, 4, 4i, 4i
21. 2, 2, 2 2, 2 2
22.  2, 2, 3, 3
40. The turning points of a polynomial function that models 9  9i 3
 9  9i 3

a set of data can indicate fluctuations that may repeat. 23. 9, 
2
, 
2
Answers should include the following. 24. 8, 4  4i 3 , 4  4i 3
• Polynomial equations best model data that contain 25. 81, 625
turning points, rather than a constant increase or
343, 64

Additional Answers for Chapter 7

26.
decrease like linear equations.
27. 225, 16
• To determine when the percentage of foreign-born
citizens was at its highest, look for rel. max. of the 28. 400
graph, which is at t  5. The lowest percentage is 38. Answers should include the following.
found at t  75, the rel. min. of the graph. • Solve the cubic equation 4x3  (164x2)  1600x 
3600 in order to determine the dimensions of the cut
Page 359, Follow-Up of Lesson 7-2 square if the desired volume is 3600 in3. Solutions
Graphing Calculator Investigation 31  601

are 10 in. and 
2
in.
1.
• There can be more than one square cut to produce
the same volume because the height of the box is not
specified and 3600 has a variety of different factors.
41. x f(x) f (x )
2 21
[1930, 2010] scl: 10 by [0, 200] scl: 20
1 1
0 5
1 3
O x
2 1
3 1
4 9 f (x)  x 3
4x 2  x  5

[1930, 2010] scl: 10 by [0, 200] scl: 20 5 35

42. x f(x) f (x )
1 15
0 3
1 1 O x
2 3
[1930, 2010] scl: 10 by [0, 200] scl: 20 3 3
4 25
f (x)  x 4
6x 3  10x 2
x
3

Chapter 7 Additional Answers 407D

 Page 364, Practice Quiz 1 Pages 393–394, Lesson 7-8
3. Sample answer: maximum at x  2, minimum at 7. f (x ) 8. g (x )
4
x  0.5 g (x )  3x  1
f (x) 
x
f (x ) 2 f
1(x ) 
x 2
8

4
4
2 O 2 4x
4
2 O 2 4x

2
2

4
2 O 2 4x
g
1(x)  13 x
13

4
4

8 9. y y  1x  5
12 2
f (x)  x  2x 2
4x
6
3

Pages 369–370, Lesson 7-4 4

32. 8 1 4 29 24
8 32 24 O 4 8 12 x

1
1 4 3 0
4 y  2x
10

37. 5 1 14 69 140 100

20. y 21. g (x )
5 45 120 100 4 4
x 
3 g
1(x ) 
12 x
Additional Answers for Chapter 7

1 9 24 20 0
2 2
x
Pages 375–377, Lesson 7-5
4
2 O 2 4x
4
2 O 2 4
41a. f (x ) 41b. f (x )
2
2

4 y 
3 g (x ) 
2x

O x O x 22. 23.
f (x ) g (x )
4 4

1
f (x)  x  5
2 2
x g (x )  x  4

41c. f (x )
4
2 O 2 4
4
2 O 2 4x

1

2 g (x)  x
2

4
f (x)  x
5

4
4
O x
24. f (x ) 25. y
4 4
y
1 
12x
12
2 f (x)  3x  3

51. If the equation models the level of a medication in a

4
2 O 2 4x
4
2 O 2 4x
patient’s bloodstream, a doctor can use the roots of the
equation to determine how often the patient should take
2
2
the medication to maintain the necessary concentra- f
1(x)  13 x
1 y 
2x
1

4
4
tion in the body. Answers should include the following.
• A graph of this equation reveals that only the first
26. y 27. f (x )
positive real root of the equation, 5, has meaning for 4 4

1
this situation, since the next positive real root occurs y  3x
after the medication level in the bloodstream has 2 2
dropped below 0 mg. Thus according to this model,
after 5 hours there is no significant amount of
4
2 O 2 4x
4
2 O 2 4x
medicine left in the bloodstream.
2
2 f (x)  58 x
• The patient should not go more than 5 hours before y  13 x

4
4
taking their next dose of medication.
f
1(x)  85 x

407E Chapter 7 Additional Answers

 5 35 Pages 397–399, Lesson 7-9
28. f (x ) 29. f 1(x) 
x 

8 4 4 4. y 5. y
7 f (x)  1 x  4 8 8
3 f (x)
1  54 x  35
4 7 7
6
5 f (x ) 6 6
5 5

40
30
20
10 O x
3 4 4

10 3 3
2 y  4x
1 2 2

20 y  x  2 1
1
O 1 2 3 4 5 6 7 8x

30 O 1 2 3 4 5 6 7 8x O 1 2 3 4 5 6 7 8x
f
1(x )  3x
12
f (x)  45 x
7

40 6. y 7. y
8
4 7
3 8 4 y  3
x 6
30. g 1(x)  3x 
31. f 1(x) 
x 
2 5
2 7 7
4
g (x ) f (x ) x
4 4 3 y  x
1  3
O 4 8 12 2
f (x)  87 x 2 47

1

g (x)  2x 6 32
2 1
O 1 2 3 4 5 6 7 8x

4
2 O 2 4x
4
2 O 2 4x

2
2 8. y 9. y
f (x)  7x 8
4 8 8

Additional Answers for Chapter 7

4
4 7
6 6
5 y  2x  4
g (x)  3x
32

1
y x
4  1
4 4
3
2 2
Page 396, Lesson 7-9 1
Graphing Calculator Investigation O 1 2 3 4 5 6 7 8x
2 O 2 4 6x
1.
10. y 11. y
4 4 y  x  2
1
3 3
2 y  3
5x  1 2
1 1

1 O 1 2 3 4 5 6 7x
2 O 1 2 3 4 5 6x
[
2, 8] scl: 1 by [
4, 6] scl: 1
2
2

3
3
2.
4
4

14. y 15. y
8
7 O 1 2 3 4 5 6 7 8x

1
6
2
5
3
[
10, 10] scl: 1 by [
10, 10] scl: 1 4
4
3
5
3. 2 y  3x
6
1
7 y 
5x
O 1 2 3 4 5 6 7 8x
8

16. y 17. y
8
O 1 2 3 4 5 6 7 8x 7

1
[
10, 10] scl: 1 by [
10, 10] scl: 1
2 6

3 5

4 4
y 
4x 3 y  1 x

5 2

6 2

7 1

8 O 1 2 3 4 5 6 7 8x

Chapter 7 Additional Answers 407F

 18. y 19. y 30. y 31. y
8 8 8 8
7 7 7
6 6 6 6
5 5 5
y  x  2 4
4 4 4
3 y  x
7 3 y  x
3  4 3 y  6x
2  1
2 2 2 2
1 1 1

2 O 2 4 6x O 1 2 3 4 5 6 7 8x O 1 2 3 4 5 6 7 8x O 1 2 3 4 5 6 7 8x

20. y 21. y
8
O 1 2 3 4 5 6 7 8x Pages 400–404, Chapter 7 Study Guide and Review

1 7

2 x4 x  3
6
48. f 1(x) 
49. f 1(x) 


3 5 3 2

4 4 f (x ) f (x )

5 3 4
y 
2x  1 -
1
(x)  x  4 f (x ) 
2x
3

6 2 y  5x
3 f 3

7 1 2
1
f (x ) 
x 2
3

8
O 1 2 3 4 5 6 7 8x

4
2 O 2 4x
4
2 O 2 4x
22. 23.
2
2
y y
4 8
4
4
Additional Answers for Chapter 7

2 6 f (x)  3x
4

O x y  5
x  4 2x  1

6
4
2 2
4
50. g 1(x)  3x  6 51. f 1(x) 

3

2 2 f (x )
g (x )  13 x  2 4
y  x  6
3
4
O
g (x ) f (x) 
3x2 1

4
2 2 4x 4 2

24. y 25. y 2
8 8
4
2 O 2 4x
7
2
6 y  23
4x  3 6
4
2 O 2 4x
f (x)  2x
1

1

3
5
2
4 y  3x
6  4 4
3
4
2 2
1 g
1(x )  3x
6
O 1 2 3 4 5 6 7 8x
3
2
1 O x
1 3
52. y 1(x)  x 53. y 1(x)  
x 

2 2
26. y 27. y y y
8 4 4
O 1 2 3 4 5 6 7 8x

2 y  (2x  3)2

4 6 2 2

6 y  x2

8 4 x
y 
6x y  x  5

10
4
2 O 2 4x
4
2 O 2 4

12 2
2
2

14
O
1

16 y  x y   12 x
32

4
2 2 x
4

28. y 29. y
8 8
7
6 6
y  2x  8 5
4 4
3
2 2 y  5x
8
1
O

4
2 2 4x O 1 2 3 4 5 6 7 8x

407G Chapter 7 Additional Answers

Page 405, Chapter 7 Practice Test 6a. f (x ) 6b. between 4 and 3
8
4a. g (x ) 4b. between 5 and 4, 6c. Sample answer:
zero at x  2, 4 rel. max. at x  2.3,
4 between 0 and 1 rel. min. at x  0.3
4c. Sample answer:
4
2 O 2 4x

4
2 O 2 4x rel. max. at x  3.5,
4

4
rel. min. at x  0.5
x 3  3x 2
2x  1
f (x ) 
8
3 2
g (x )  x
8
 6x  6x
4

7a. g (x ) 7b. between 3 and 2,

8
5a. 5b. between 4 and 3, between 1 and 0,
h (x )
between 3 and 2, 4 between 1 and 2,
4 zero at x  0 between 2 and 3
5c. Sample answer:
4
2 O 2 4x 7c. Sample answer:

4
2 4x rel. max. at x  1, rel. max. at x  0.6,
O 2
4
rel. min. at x  3 rel. min. at x  1.5

4
and at x  0
8 and at x  2.4
4 3 2 g (x)  x 4
2x 3
6x 2 8x  5
h (x)  x  6x  8x
x

Additional Answers for Chapter 7

Chapter 7 Additional Answers 407H