Notes Polynomial

and Radical
Introduction Equations and
and
In this unit, students extend their
knowledge of first-degree equa-
tions and their graphs to radical
Equations that model
real-world data allow
you to make predictions
Inequalities
equations and inequalities. Then about the future.
they graph quadratic functions In this unit, you will
and solve quadratic equations Chapter 5
learn about nonlinear
and inequalities by various Polynomials
equations, including
methods, including completing polynomial and Chapter 6
the square and using the radical equations, Quadratic Functions and
Quadratic Formula.
and inequalities. Inequalities
The unit concludes with methods
Chapter 7
for evaluating polynomial func-
Polynomial Functions
tions, including the Remainder
and Factor Theorems. Students
graph polynomial functions and
investigate their roots and zeros.
Finally, they study the compo-
sition of two functions, and then
find the inverse of a function.

Assessment Options
Unit 2 Test Pages 449–450
of the Chapter 7 Resource Masters
may be used as a test or review
for Unit 2. This assessment con-
tains both multiple-choice and
short answer items.

TestCheck and
Worksheet Builder
This CD-ROM can be used to
create additional unit tests and
review worksheets.

218 Unit 2 Polynomial and Radical Equations and Inequalities

218 Unit 2 Polynomial and Radical Equations and Inequalities

 Teaching
Suggestions

Have students study the

USA TODAY Snapshot®.
• Have students make conjec-
tures about why a quadratic
or polynomial model may be
better than a linear one for
modeling population data.
• According to the given data,
what urban area has a popu-
lation nearly as great as that
of New York and Los
Angeles combined? Tokyo
Additional USA TODAY
Snapshots® appearing in Unit 2:
Chapter 5 Hanging on to the
old buggy (p. 228)
Chapter 6 More Americans
study abroad
(p. 328)
Population Explosion
Chapter 7 Digital book sales
The United Nations estimated that the world’s expected to grow
USA TODAY Snapshots®
population reached 6 billion in 1999. The population (p. 368)
had doubled in about 40 years and gained 1 billion Tokyo leads population giants
people in just 12 years. Assuming middle-range birth The 10 most populous urban areas in the world:

and death trends, world population is expected to

exceed 9 billion by 2050, with most of the increase in
(Millions)
countries that are less economically developed. In this Tokyo 26.4
project, you will use quadratic and polynomial Mexico City 18.1
Bombay, India 18.1
mathematical models that will help you to project
Sao Paulo, Brazil 17.8
future populations. New York 16.6
Lagos, Nigeria 13.4
Log on to www.algebra2.com/webquest. Los Angeles 13.1
Begin your WebQuest by reading the Task. Shanghai, China 12.9
Calcutta, India 12.9
Then continue working Buenos Aires, Argentina
Lesson 5-1 6-6 7-4 12.1
on your WebQuest as Page 227 326 369 Source: United Nations By Bob Laird, USA TODAY
you study Unit 2.
Unit 2 Polynomial and Radical Equations and Inequalities 219

Internet Project
A WebQuest is an online project in which students do research on the Internet,
gather data, and make presentations using word processing, graphing,
page-making, or presentation software. In each chapter, students advance to
the next step in their WebQuest. At the end of Chapter 7, the project
culminates with a presentation of their findings.
Teaching suggestions and sample answers are available in the WebQuest and
Project Resources.

Unit 2 Polynomial and Radical Equations and Inequalities 219

 Polynomials
Chapter Overview and Pacing

PACING (days)
Regular Block
LESSON OBJECTIVES Basic/ Basic/
Average Advanced Average Advanced
Monomials (pp. 222–228) 1 1 0.5 0.5
• Multiply and divide monomials.
• Use expressions written in scientific notation.
Polynomials (pp. 229–232) 1 1 0.5 0.5
• Add and subtract polynomials.
• Multiply polynomials.
Dividing Polynomials (pp. 233–238) 1 1 0.5 0.5
• Divide polynomials using long division.
• Divide polynomials using synthetic division.
Factoring Polynomials (pp. 239–244) 2 2 1 1
• Factor polynomials.
• Simplify polynomial quotients by factoring.
Roots of Real Numbers (pp. 245–249) 1 1 0.5 0.5
• Simplify radicals.
• Use a calculator to approximate radicals.
Radical Expressions (pp. 250–256) 2 2 1 1
• Simplify radical expressions.
• Add, subtract, multiply, and divide radical expressions.
Rational Exponents (pp. 257–262) 2 2 1 1
• Write expressions with rational exponents in radical form, and vice versa.
• Simplify expressions in exponential or radical form.
Radical Equations and Inequalities (pp. 263–269) 2 2 1 1
• Solve equations containing radicals. (with 4-8
• Solve inequalities containing radicals. Follow-Up)
Follow-Up: Solving Radical Equations and Inequalities by Graphing
Complex Numbers (pp. 270–275) 2 2 1 1
• Add and subtract complex numbers.
• Multiply and divide complex numbers.
Study Guide and Practice Test (pp. 276–281) 1 1 0.5 0.5
Standardized Test Practice (pp. 282–283)
Chapter Assessment 1 1 0.5 0.5
TOTAL 16 16 8 8

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

220A Chapter 5 Polynomials

 Timesaving Tools
™

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

CHAPTER 5 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
239–240 241–242 243 244 SM 109–114 5-1 5-1

245–246 247–248 249 250 SC 9 5-2 5-2 algebra tiles

251–252 253–254 255 256 307 5-3 5-3

257–258 259–260 261 262 GCS 35 5-4 5-4 8 algebra tiles, graphing calculator

263–264 265–266 267 268 307, 309 5-5 5-5 index cards, string, small weights

269–270 271–272 273 274 5-6 5-6

275–276 277–278 279 280 308 GCS 36 5-7 5-7

281–282 283–284 285 286 5-8 5-8 9 (Follow-Up: graphing calculator)

287–288 289–290 291 292 308 SC 10 5-9 5-9 grid paper

293–306,
310–312

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

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

Chapter 5 Polynomials 220B

 Mathematical Connections
and Background
Continuity of Instruction Monomials
Throughout this chapter students are introduced
to new symbols and new notation. Each time, the new
symbols are related to each other and to familiar ideas. In
this lesson the new symbol is a negative exponent. The
Prior Knowledge familiar idea is simplifying an expression, and to simplify
Students are familiar with the coefficients, a monomial means to write an equivalent expression with-
out negative exponents and without parentheses. The
variables, and positive integer exponents that
rules for writing equivalent expressions include the defi-
make up monomials, and they have worked nition of negative exponents and properties of exponents.
with radicals as square roots. They are familiar The lesson introduces the terms coefficient and
with the basic arithmetic operations (addition, degree when discussing monomials. It also explores writ-
subtraction, multiplication, division) that will ing and operating on numbers written in scientific nota-
be applied in new situations in this tion, and using dimensional analysis to compute with
units of measure.
chapter.
Polynomials
In this lesson students find the sum or difference
of several monomials, which is called a polynomial. The
familiar ideas in this lesson are the operations of addition,
This Chapter subtraction, and multiplication as applied to polynomials.
Students learn how to apply the basic arith- To add polynomials means to rewrite an indicated sum of
metic operations to polynomials, radical ex- polynomials as a sum of terms and then, by combining
pressions, and complex numbers. They explore like terms, to rewrite that sum as a single polynomial.
To subtract polynomials means to use the Distributive
factoring polynomials and solving radical
Property to rewrite the subtraction as a sum of terms, and
equations and inequalities. They learn how then to combine like terms. To multiply a polynomial by
to write equivalent statements by using the a monomial means to use the Distributive Property to
Distributive Property, properties of exponents, rewrite the product as a single polynomial.
or properties of radicals. They solve equations To multiply two binomials, the lesson shows how
two applications of the Distributive Property result in
and inequalities involving polynomials,
finding the products of the First, Outer, Inner, and Last
radicals, or complex numbers. pairs of terms of the binomials. The two-time application
of the Distributive Property is called the FOIL method.

Dividing Polynomials
This lesson explores polynomial division; with the
previous lesson, the four basic arithmetic operations are
Future Connections interpreted for polynomials. Dividing by a monomial uses
As early as the next two chapters, students the Distributive Property. Dividing a polynomial by a
will factor polynomials to solve quadratic binomial (or by any polynomial) uses a process and for-
equations, use complex numbers to express mat analogous to the long division algorithm for whole
numbers. The student follows the four steps of “divide,
the solution to equations or inequalities,
multiply, subtract, bring down”; the steps are repeated
and relate complex numbers and roots of until there are no more terms in the dividend.
polynomials. In their exploration of complex The lesson also introduces an abbreviated form,
numbers, they will greatly expand on this called synthetic division, that records and manipulates
chapter’s brief introduction to the complex just the coefficients of the polynomial terms. The divisor
coordinate plane. must be a binomial of degree 1, the terms of the dividend
must be in descending order, using zeros to represent any
missing terms, and the polynomial quotient must be
written so that the leading coefficient of the divisor is 1.
220C Chapter 5 Polynomials
 Factoring Polynomials Radical Equations and
In factoring, a polynomial of degree two or Inequalities
more is rewritten as a product of polynomials each This lesson deals with the familiar skills of
having a lesser degree. Taking out a common factor solving equations and inequalities; the new concept is
uses the Distributive Property; factoring by grouping that the equations contain a variable inside a radical.
uses two applications of the Distributive Property. No new properties are needed to solve these equa-
Binomials written in the form of the difference tions or inequalities; equivalent equations or inequali-
of two squares or as the sum or difference of two ties are written until the variable is isolated on one
cubes can be rewritten as a product of two factors, side. At least once in the solution, both sides of the
and a perfect square trinomial can be rewritten as the equation or inequality are raised to a power in order
square of a binomial. Some other trinomials can be to remove a radical symbol. A most-important idea in
factored as the product of two binomials. the lesson is that sometimes this process of raising
Students reduce quotients of polynomials by both sides to a power does not produce an equivalent
removing common factors in the numerator and
statement. For example, it is clear that x  5 has
denominator. A record is kept of values of variables
no real number solution. Squaring both sides results
that would imply division by zero.
in x  25; the two statements are not equivalent and
x  25 is not a solution to the original equation. Rais-
Roots of Real Numbers ing both sides to a power can introduce an extraneous
This lesson begins with the familiar idea of a solution, which is an apparent solution that will not
square root: a is a square root of b if a2  b. Then two satisfy the original equation or inequality.
new ideas are introduced. One idea is to use the same
kind of definition to introduce the nth root of a num- Complex Numbers
ber: a is an nth root of b if an  b. The second new idea
This lesson introduces not simply a new sym-
is to introduce the symbols for principal roots. The
bol but a new set of numbers that are not part of the
principal square root is always a nonnegative number.
real number system. The new symbol is i, and a new
The value of a principal nth root depends on the sign
of the radicand and whether its index is even or odd. rule is that an expression such as  5 can be rewrit-
The lesson explores how to use absolute value symbols ten as the equivalent expression i5 . The complex
to simplify nth roots. number a  bi can be treated as if it is a binomial, and
operations on complex numbers follow the properties
for adding, subtracting, multiplying, and dividing
Radical Expressions binomials, with one exception. That exception is to re-
This lesson explains that “simplifying,” as it place i 2 with 1 whenever i 2 appears in an expression.
pertains to radical expressions, takes into account the
index of the radical and the form of the radicand. The
lesson also presents some rules for writing equivalent
radical expressions, and students apply the rules as
they add, subtract, multiply, and divide radicals.
www.algebra2.com/key_concepts
Rational Exponents
Additional mathematical information and teaching notes
The new symbol introduced in this lesson is a are available in Glencoe’s Algebra 2 Key Concepts:
fraction used as an exponent. The rules for writing Mathematical Background and Teaching Notes,
equivalent expressions for rational exponents include which is available at www.algebra2.com/key_concepts.
properties that describe how to translate between The lessons appropriate for this chapter are as follows.
radical form and exponential form. Those rules include
dealing with rational exponents that are unit fractions, • Multiplying Monomials (Lesson 22)
either positive or negative. The rules for dealing with • Dividing Monomials (Lesson 23)
a base raised to a fractional exponent require that the • Adding and Subtracting Polynomials (Lesson 24)
denominator of the fraction is a positive integer and • Multiplying a Polynomial by a Monomial (Lesson 25)
take into account the sign of the radicand and whether • Multiplying Polynomials (Lesson 26)
its index is even or odd.

Chapter 5 Polynomials 220D

 and Assessment

Type Student Edition Teacher Resources Technology/Internet

Ongoing Prerequisite Skills, pp. 221, 228, 5-Minute Check Transparencies Alge2PASS: Tutorial Plus
INTERVENTION

232, 238, 244, 249, 256, 262, Quizzes, CRM pp. 307–308 www.algebra2.com/self_check_quiz
267 Mid-Chapter Test, CRM p. 309 www.algebra2.com/extra_examples
Practice Quiz 1, p. 238 Study Guide and Intervention, CRM pp. 239–240,
Practice Quiz 2, p. 256 245–246, 251–252, 257–258, 263–264, 269–270,
275–276, 281–282, 287–288
Mixed pp. 228, 232, 238, 244, 249, Cumulative Review, CRM p. 310
Review 256, 262, 267, 275
Error Find the Error, pp. 226, 236 Find the Error, TWE pp. 226, 236
Analysis Unlocking Misconceptions, TWE pp. 223, 235, 244,
246, 253, 258
Tips for New Teachers, TWE pp. 228, 238, 244, 246,
256, 262, 267, 275
Standardized pp. 228, 232, 234, 236, 238, TWE p. 234 Standardized Test Practice
Test Practice 244, 249, 255, 262, 267, 275, Standardized Test Practice, CRM pp. 311–312 CD-ROM
281, 282–283 www.algebra2.com/
standardized_test
Open-Ended Writing in Math, pp. 227, 232, Modeling: TWE pp. 244, 249
Assessment 238, 243, 249, 255, 262, 267, Speaking: TWE pp. 228, 256, 262, 275
275 Writing: TWE pp. 232, 238, 267
Open Ended, pp. 226, 231, 236, Open-Ended Assessment, CRM p. 305
ASSESSMENT

242, 247, 254, 260, 265, 273

Chapter Study Guide, pp. 276–280 Multiple-Choice Tests (Forms 1, 2A, 2B), TestCheck and Worksheet Builder
Assessment Practice Test, p. 281 CRM pp. 293–298 (see below)
Free-Response Tests (Forms 2C, 2D, 3), MindJogger Videoquizzes
CRM pp. 299–304 www.algebra2.com/
Vocabulary Test/Review, CRM p. 306 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.

220E Chapter 5 Polynomials

 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. 221
5-4 8 Factoring Expressions II • Concept Check questions require students to verbalize
5-8 9 Solving Radical Equations and write about what they have learned in the lesson.
(pp. 226, 231, 236, 242, 247, 254, 260, 265, 273, 280)
ALEKS is an online mathematics learning system that • Writing in Math questions in every lesson, pp. 227, 232,
adapts assessment and tutoring to the student’s needs. 238, 243, 249, 255, 262, 267, 275
Subscribe at www.k12aleks.com. • Reading Study Tip, pp. 229, 246, 252, 270, 271, 273
• WebQuest, p. 227

Intervention at Home Teacher Wraparound Edition

• Foldables Study Organizer, pp. 221, 276
Log on for student study help. • Study Notebook suggestions, pp. 226, 230, 236, 242,
• For each lesson in the Student Edition, there are Extra 247, 254, 260, 265, 273
Examples and Self-Check Quizzes. • Modeling activities, pp. 244, 249
www.algebra2.com/extra_examples • Speaking activities, pp. 228, 256, 262, 275
www.algebra2.com/self_check_quiz • Writing activities, pp. 232, 238, 267
• For chapter review, there is vocabulary review, test • Differentiated Instruction, (Verbal/Linguistic), p. 271
practice, and standardized test practice. • ELL Resources, pp. 220, 227, 231, 237, 243, 248,
www.algebra2.com/vocabulary_review 255, 261, 266, 271, 274, 276
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 5 Resource
Assessment, see pp. T8–T11. Masters, pp. vii-viii)
• Reading to Learn Mathematics master for each lesson
(Chapter 5 Resource Masters, pp. 243, 249, 255, 261,
267, 273, 279, 285, 291)
• 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 5 Polynomials 220F
 Notes
Polynomials
Have students read over the list
of objectives and make a list of
any words with which they are
not familiar. • Lessons 5-1 through 5-4 Add, subtract, multiply,
Key Vocabulary
divide, and factor polynomials. • scientific notation (p. 225)
• Lessons 5-5 through 5-8 Simplify and solve • polynomial (p. 229)
equations involving roots, radicals, and rational • FOIL method (p. 230)
exponents.
• synthetic division
• Lesson 5-9 Perform operations with complex (p. 234)
Point out to students that this is numbers.
• complex number
only one of many reasons why (p. 271)
each objective is important.
Others are provided in the
introduction to each lesson.
Many formulas involve polynomials and/or square
roots. For example, equations involving speeds or
velocities of objects are often written with square
roots. You can use such an equation to find the
velocity of a roller coaster. You will use an equation
relating the velocity of a roller coaster and the height of a
hill in Lesson 5-6.

NCTM Local
Lesson Standards Objectives
5-1 1, 2, 6, 7, 8, 9,
10
5-2 1, 2, 6, 8, 9, 10
5-3 1, 2, 6, 7, 8, 9
5-4 1, 2, 3, 6, 8, 9,
10
5-5 1, 2, 6, 7, 8, 9
5-6 1, 2, 6, 7, 8, 9,
10
5-7 1, 2, 6, 8, 9
5-8 1, 2, 6, 8, 9, 10
220 Chapter 5 Polynomials
5-8 1, 2, 10
Follow-Up
5-9 1, 2, 3, 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 5 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 5 test.
10=Representation
220 Chapter 5 Polynomials
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 5.
beginning Chapter 5. Page
For Lessons 5-2 and 5-9 Rewrite Differences as Sums
references are included for
additional student help.
Rewrite each difference as a sum.
1. 2  7 2  (7) 2. 6  11 6  (11) 3. x  y x  (y ) Prerequisite Skills in the Getting
4. 8  2x 8  (2x) 5. 2xy  6yz 2xy  (6yz) 6. 6a2b  12b2c Ready for the Next Lesson section
6a 2b  (12b 2c)
at the end of each exercise set
For Lesson 5-2 Distributive Property review a skill needed in the next
Use the Distributive Property to rewrite each expression without parentheses. lesson.
(For review, see Lesson 1-2.) 7. 8x 3  2x  6
7. 2(4x3  x  3) 8. 1(x  2) x  2 9. 1(x  3) x  3 For Prerequisite
1 3 2 4 Lesson Skill
10. 3(2x4  5x2  2) 11. (3a  2) a  1 12. (2  6z)   4z
2 2 3 3
6x 4  15x 2  6 5-2 Distributive Property (p. 228)
For Lessons 5-5 and 5-9 Classify Numbers 5-3 Properties of Exponents
Find the value of each expression. Then name the sets of numbers to which each value
(p. 232)
belongs. (For review, see Lesson 1-2.) 13–18. See margin. 5-5 Rational and Irrational
13. 2.6  3.7 14. 18  (3) 15. 23  32 Numbers (p. 244)
18  14 5-6 Multiplying Binomials (p. 249)
16. 
41 17.  18. 34
8
5-8 Multiplying Radicals (p. 262)
5-9 Binomials (p. 267)

Make this Foldable to record information about polynomials. Answers

Begin with four sheets of grid paper.
13. 6.3; reals, rationals
Fold and Cut Fold and Label 14. 6; reals, rationals, integers
First Sheets Second Sheets 15. 17; reals, rationals, integers,
Insert first sheets whole numbers, natural numbers
through second sheets
and align folds. Label 16. 5; reals, irrationals
pages with lesson
numbers. 17. 4; reals, rationals, integers,
whole numbers, natural numbers
18. 6; reals, rationals, integers,
Fold in half along whole numbers, natural numbers
the width. On the first two
sheets, cut along the fold at
the ends. On the second two
sheets, cut in the center of
the fold as shown.

Reading and Writing As you read and study the chapter, fill the journal with
notes, diagrams, and examples for polynomials.

Chapter 5 Polynomials 221

TM

Organization of Data and Journal Writing When labeling the

pages for the lessons, combine Lessons 5-1 and 5-2 on the same
page and Lessons 5-8 and 5-9 on the same page. Use extra pages
For more information for vocabulary lists and applications. Writer’s journals can also be used
about Foldables, see to record the direction and progress of learning, to describe positive
Teaching Mathematics and negative experiences during learning, to write about personal
with Foldables. associations and experiences while learning, and to list examples of
ways in which new knowledge has or will be used in their daily life.

Chapter 5 Polynomials 221

 Lesson Monomials
Notes

• Multiply and divide monomials.

1 Focus • Use expressions written in scientific notation.

U.S. Public Debt
Vocabulary is scientific notation useful
5-Minute Check in economics?

$
• monomial

0
$
$
0
$

00, 00
Transparency 5-1 Use as a

$
00, 00
0, 000
$
• constant

$
Economists often deal with very large

$
$

, 00
00
$
quiz or a review of Chapter 4. • coefficient

, 200, 0
numbers. For example, the table shows

, 300, 0
, 000, 0

0, 000

00, 00
Debt ($)
• degree the U.S. public debt for several years in

5, 674
16, 10

284, 1
Mathematical Background notes

3, 233
1, 200
• power the last century. Such numbers, written in

$
$
• simplify

$
are available for this lesson on standard notation, are difficult to work

$
$

$
$
$
$
$
p. 220C. • standard notation with because they contain so many digits. 1900 1930 1960 1990 2000
• scientific notation Scientific notation uses powers of ten to Year
• dimensional analysis make very large or very small numbers
is scientific notation more manageable. Source: U.S. Department of the Treasury
useful in economics?
Ask students: MONOMIALS A monomial is an expression that is a number, a variable, or the
• What are the powers of ten? product of a number and one or more variables. Monomials cannot contain variables
in denominators, variables with exponents that are negative, or variables under radicals.
…, 103, 102, 101, 100, 101,
102, 103, … Monomials Not Monomials
1 1
x, x  8, a1
3
• What are some other fields that 5b, w, 23, x2, x3y4  ,
3 n4
use scientific notation for very
Constants are monomials that contain no variables, like 23 or 1. The numerical
large or very small numbers? factor of a monomial is the coefficient of the variable(s). For example, the coefficient
astronomy, biology, computer of m in 6m is 6. The degree of a monomial is the sum of the exponents of its
science variables. For example, the degree of 12g7h4 is 7  4 or 11. The degree of a constant is 0.
A power is an expression of the form xn. The word power is also used to refer to
the exponent itself. Negative exponents are a way of expressing the multiplicative
2 Teach 1
inverse of a number. For example, 2 can be written as x2. Note that an expression
such as x2 is not a monomial. Why?
x

MONOMIALS
Negative Exponents
In-Class Example Power
Point®
• Words 1
For any real number a  0 and any integer n, an  
1
  an.
n and n a a
• Examples 23  213 and 1
  b8
b8
Teaching Tip Help students
think carefully about the meaning
of exponents by asking them to To simplify an expression containing powers means to rewrite the expression
read this expression aloud without parentheses or negative exponents.
correctly. If students read x3 as
“x three,” instead of correctly Example 1 Simplify Expressions with Multiplication
saying “x cubed” or “x to the Simplify (3x3y2)(4x2y4).
third (power),” they are apt to
confuse x3 with 3x. (3x3y2)(4x2y4)  (3  x  x  x  y  y)(4  x  x  y  y  y  y) Definition of exponents
 3(4)  x  x  x  x  x  y  y  y  y  y  y Commutative Property
1 Simplify (2a3b)(5ab4).  12x5y6 Definition of exponents
10a 4b 5
222 Chapter 5 Polynomials

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 5 Resource Masters Science and Mathematics Lab Manual, 5-Minute Check Transparency 5-1
• Study Guide and Intervention, pp. 239–240 pp. 109–114 Answer Key Transparencies
• Skills Practice, p. 241
• Practice, p. 242 Technology
• Reading to Learn Mathematics, p. 243 Interactive Chalkboard
• Enrichment, p. 244
 Example 1 suggests the following property of exponents.
In-Class Example Power
Point®
2
Product of Powers s
2 Simplify 10 . Assume that
s
• Words For any real number a and integers m and n, am  an  am  n. 1
• Examples 42  49  411 and b3  b5  b8 s  0. 8
s
Teaching Tip When discussing
To multiply powers of the same variable, add the exponents. Knowing this, In-Class Example 2, if any stu-
it seems reasonable to expect that when dividing powers, you would subtract dents get the incorrect answer
x9 1
exponents. Consider 5 .  , lead them to understand
x
s5
1 1 1 1 1
x9 x  x  x  x  x  x  x  x  x that they divided the exponents
   Remember that x ≠ 0. instead of subtracting them as a
x5 x  x  x  x  x
1 1 1 1 1 method for dividing the two
xxxx Simplify. expressions.
 x4 Definition of exponents
It appears that our conjecture is true. To divide powers of the same base, you
subtract exponents.
Interactive
Quotient of Powers Chalkboard
PowerPoint®
m  n. am
• Words For any real number a  0, and integers m and n, 
n a Presentations
a
3 7
• Examples 55  53  1 or 52 and xx3  x7  3 or x 4
This CD-ROM is a customizable
Microsoft® PowerPoint®
presentation that includes:
Example 2 Simplify Expressions with Division
• Step-by-step, dynamic solutions of
p3
Simplify 8 . Assume that p  0. each In-Class Example from the
p
p3 Teacher Wraparound Edition
  p3  8 Subtract exponents.
p8 • Additional, Your Turn exercises for
1
 p5 or 5 Remember that a simplified expression cannot contain negative exponents. each example
p
1 1 1 • The 5-Minute Check Transparencies
p3 p  p  p
CHECK    Definition of exponents • Hot links to Glencoe Online
p8 p  p  p  p  p  p  p  p
1 1 1
Study Tools
1
 5 Simplify.
p

You can use the Quotient of Powers property and the definition of exponents
y4
to simplify 4 , if y  0.
y
Method 1 Method 2
1 1 1 1
y4 y4 y  y  y  y
4  y4  4 Quotient of Powers 4   Definition of exponents
y y y  y  y  y
1 1 1 1

 y0 Subtract. 1 Divide.

In order to make the results of these two methods consistent, we define y 0  1, where
y  0. In other words, any nonzero number raised to the zero power is equal to 1.
Notice that 00 is undefined.

www.algebra2.com/extra_examples Lesson 5-1 Monomials 223

Unlocking Misconceptions
• Correcting Errors Encourage students to analyze the error or errors
they made when they get an incorrect answer. Stress that students
should use errors as an opportunity to clarify their thinking.
• Using Definitions Suggest that students return to the basic defini-
tions for exponents rather than just memorizing rules. For example,
they can derive the rule for multiplying quantities such as x2  x3 by
rewriting the problem as x  x  x  x  x.

Lesson 5-1 Monomials 223

 The properties we have presented can be used to verify the properties of powers
In-Class Examples Power
Point® that are listed below.

3 Simplify each expression. Properties of Powers

a. (b2)4 b8 • Words Suppose a and b are real numbers and m • Examples
and n are integers. Then the following
b. (3c2 d 5)3 27c 6d 15 properties hold.
Power of a Power: (am)n  amn (a2)3  a6
 2a
b 
5
32a 5
c. 2  10 Power of a Product: (ab)m  ambm (xy)2  x2y2
b
an
x 4 81
d.   
a n
  ba a3
3
Power of a Quotient:    , b  0 and  3
 4
b bn b
3 x n bn y4
a

b a  
b n
  or 
an  
, a  0, b  0  
x
y
4
 4
x
Teaching Tip Encourage
students to begin each step of a
simplification by naming the
operation they are about to Example 3 Simplify Expressions with Powers
perform, such as finding the
power of a power. Simplify each expression.
a. (a3)6 b. (2p3s2)5
3a5y 5
4 Simplify 
a6yb4 
. 
243

a 5yb 20
(a3)6  a3(6) Power of a power (2p3s2)5  (2)5  (p3)5  (s2)5
 a18  32p15s10 Power of a power
3x 4 3
c.  d. 
a
y 4
3x 4 (3x)4 a 3 4 3
Concept Check   
y y4
Power of a quotient  

4
 
a   Power of a quotient

Monomials Have students write (3)4x4

  Power of a product
43
 3 Power of a quotient
4 y a
their own summary of the prop-
erties of exponents, such as 81x4 64
 
4 (3)4  81  
3 43  64
y a
“to multiply expressions with
exponents, you add the expo-
nents; to divide, you subtract the Study Tip
exponents” and so on. With complicated expressions, you often have a choice of which way to start
Simplified
Expressions simplifying.
A monomial expression is
in simplified form when: Example 4 Simplify Expressions Using Several Properties
• there are no powers of
2x3n 4
powers, Simplify 2n3  .
• each base appears x y
exactly once, Method 1 Method 2
• all fractions are in
simplest form, and Raise the numerator and denominator Simplify the fraction before raising
• there are no negative to the fourth power before simplifying. to the fourth power.
exponents. 4
2x 3n 4 (2x3n) 2x 3n 4 2x3n  2n 4

x2ny3 
 2n
3 4
(x y ) 
x2ny3 
  
3  y
4 4
(2) (x3n) 2xn 4

2n 4
3 4  
3 
(x ) (y ) y
16x12n 16x4n

8n
12  12

x y y
16x12n  8n
 1
2 y
16x4n
 12

y

224 Chapter 5 Polynomials

224 Chapter 5 Polynomials

 SCIENTIFIC NOTATION The form that you usually write numbers in is SCIENTIFIC NOTATION
standard notation. A number is in scientific notation when it is in the form
a
10n, where 1  a  10 and n is an integer. Scientific notation is used to express In-Class Examples Power
very large or very small numbers. Point®

5 Express each number in

Study Tip Example 5 Express Numbers in Scientific Notation
scientific notation.
Graphing Express each number in scientific notation.
Calculators a. 4,560,000 4.56  106
a. 6,380,000
To solve scientific notation
problems on a graphing 6,380,000  6.38
1,000,000 1  6.38  10 b. 0.000092 9.2  105
calculator, use the EE  6.38
106 Write 1,000,000 as a power of 10.
function. Enter 6.38
106 6 Evaluate. Express the result
as 6.38 2nd [EE] 6. b. 0.000047 in scientific notation.
0.000047  4.7
0.00001 1  4.7  10
a. (5
103)(7
108) 3.5  1012
1 1 1
 4.7
5 0.00001   or 
100,000 105 b. (1.8
104)(4
107) 7.2  103
10
 4.7
105 Use a negative exponent.
7 BIOLOGY There are about
5
106 red blood cells in one
You can use properties of powers to multiply and divide numbers in scientific milliliter of blood. A certain
notation. blood sample contains
8.32
106 red blood cells.
Example 6 Multiply Numbers in Scientific Notation About how many milliliters
Evaluate. Express the result in scientific notation. of blood are in the sample?
about 1.66 mL
a. (4  105)(2  107)
(4
105)(2
107)  (4  2)
(105  107) Associative and Commutative Properties
 8
1012 4  2  8, 105  107  105  7 or 1012

b. (2.7  102)(3  106)

(2.7
102)(3
106)  (2.7  3)
(102  106) Associative and Commutative Properties
 8.1
104 2.7  3  8.1, 102  106  102  6 or 104

Real-world problems often involve units of measure. Performing operations with

units is known as dimensional analysis.

Example 7 Divide Numbers in Scientific Notation

ASTRONOMY After the Sun, the next-closest star to Earth is Alpha Centauri C,
which is about 4  1016 meters away. How long does it take light from Alpha
Centauri C to reach Earth? Use the information at the left.
Begin with the formula d  rt, where d is distance, r is rate, and t is time.
d
t   Solve the formula for time.
r
4
1016 m ← Distance from Alpha Centauri C to Earth
 
3.00
108 m/s ← speed of light
Astronomy 4 1016 1016
   8 Estimate: The result should be slightly greater than 
10
8
8 or 10 .
Light travels at a speed of 3.00 10 1/s
about 3.00
108 m/s. The 4 10 16
 1.33
108 s   1.33, 
3.00 108
1016  8 or 108
distance that light travels in
a year is called a light-year. It takes about 1.33
108 seconds or 4.2 years for light from Alpha Centauri C
Source: www.britannica.com to reach Earth.

Lesson 5-1 Monomials 225

Differentiated Instruction
Interpersonal Have students discuss with a partner or in a small group
the methods for multiplying and dividing monomial expressions with
exponents, and also numbers written in scientific notation. Ask them to
work together to develop a list of common errors for such problems,
and to suggest ways to correct and avoid these errors.

Lesson 5-1 Monomials 225

 3 Practice/Apply Concept Check 1. OPEN ENDED Write an example that illustrates a property of powers. Then use
multiplication or division to explain why it is true. See margin.
2. Sometimes; in 2. Determine whether xy  xz  xyz is sometimes, always, or never true. Explain.
general xy  xz  xy  z, 2a2b
Study Notebook so xy  xz  xyz when 3. FIND THE ERROR Alejandra and Kyle both simplified 
3 2 . (2ab )
Have students— y  z  yz, such as
when y  2 and z  2. Alejandra Kyle
• add the definitions/examples of 3. Alejandra; when 2a2b 2a2b 2a2b
 = (2a 2 b)(–2ab 3 ) 2  = 
the vocabulary terms to their Kyle used the Power of (–2 a b 3) –2 (–2ab3)–2 (–2)–2a(b3)–2
a Product property in = (2a 2 b)(–2) 2 a 2 (b 3 ) 2 2a2b
Vocabulary Builder worksheets for = 
his first step, he forgot 4ab–6
= (2a 2 b)2 2 a 2 b 6
Chapter 5. 2a2bb6
to put an exponent of = 
2 on a. Also, in his = 8a 4 b 7 4a
• add the information about the ab7
second step, (2)2 = 
meaning of dimensional analysis 1
2
should be , not 4.
to their notebook. 4 Who is correct? Explain your reasoning.
• include any other item(s) that they
Guided Practice Simplify. Assume that no variable equals 0.
find helpful in mastering the skills
GUIDED PRACTICE KEY 4. x2  x8 x 10 5. (2b)4 16b 4 6. (n3)3(n3)3 1
in this lesson. 30y4 81p6q5
Exercises Examples 2a b 3 6 ab4
7. 2 6y 2 8.   9. 22 9p 2q 3
4–9 1–3 5y 18a2b2 9 (3p q)
1 3 1 cd 2 9 6x6 2 1
10–12
13, 14
4
5

10. 42 126
wz w z  11.  22
3 c d 
12. 
3x 3
 
4x 6
15 6 Express each number in scientific notation.
About the Exercises… 16, 17 7
13. 421,000 4.21  105 14. 0.000862 8.62  104
Organization by Objective Evaluate. Express the result in scientific notation.
• Monomials: 18–43 8
10 1
15. (3.42
108)(1.1
105) 3.762  103 16.  2 5  10
0
• Scientific Notation: 44–60 16
10

Odd/Even Assignments Application 17. ASTRONOMY Refer to Example 7 on

Exercises 18–55 are structured page 225. The average distance from 3.84  108 m
Earth to the Moon is about 3.84
108
so that students practice the meters. How long would it take a radio
same concepts whether they signal traveling at the speed of light to
are assigned odd or even cover that distance? about 1.28 s
problems.
Alert! Exercise 59 involves
research on the Internet or
other reference materials. ★ indicates increased difficulty
Assignment Guide Practice and Apply
Basic: 19–39 odd, 45–57 odd, Homework Help Simplify. Assume that no variable equals 0.
59–84 For
Exercises
See
Examples
18. a2  a6 a 8 19. b3  b7 b 4 20. (n4)4 n16
Average: 19–57 odd, 59–84 18–35, 60 1–3
21. (z2)5 z10 22. (2x)4 16x 4 23. (2c)3 8c 3
36–39 4
Advanced: 18–58 even, 60–78 40–43 1, 2 an 2 6 y5z7 28x 4
24. 5 an 5 y z
25. 2 3 2 26. (7x3y5)(4xy3) 
(optional: 79–84) 44–49, 56, 57 5 an yz y2
50–55, 58, 59 6, 7
27. (3b3c)(7b2c2) 21b 5c 3 28. (a3b3)(ab)2 ab 29. (2r2s)3(3rs2) 24r 7s5
Extra Practice 5x3y3z4 1
30. 2x2(6y3)(2x2y) 24x 4y 4 31. 3a(5a2b)(6ab3) 90a 4b 4 32. 3
20x y7z4
4
FIND THE ERROR See page 836. 4y
Suggest that stu- 226 Chapter 5 Polynomials

dents use two steps to

simplify expressions such as
1 m factors m factors m factors
 2 by first rewriting with the



















(2)
62. (ab)m  ab  ab  …  ab  a  a  …  a  b  b  …  b  ambm
reciprocal and then squaring.
63. Economics often involves large amounts of money. Answers should include the following.
Answers • The national debt in 2000 was five trillion, six hundred seventy-four billion, two hundred
1. Sample answer: (2x 2)3  8x 6 since million or 5.6742  1012 dollars. The population was two hundred eighty-one million or
(2x 2)3  (2x 2)  (2x 2)  (2x 2)  2.81  108.
5.6742  1012
2x 2  2x 2  2x 2  • Divide the national debt by the population:  ≈ $2.0193  104 or about
2.81  108
2x  x  2x  x  2x  x  8x 6 $20,193 per person.
226 Chapter 5 Polynomials
 3a5b3c3 a2c2 2c3d(3c2d5) cd 4 12m4n8(m3n2) m4n9
33. 3  34.   35.  
NAME ______________________________________________ DATE ____________ PERIOD _____

9a b7c 3b4 30c4d2 5 36m3n 3 Study

5-1 Guide
Study andIntervention
Guide and Intervention,
3 8y 3 p. 239 (shown) and p. 240
x 2 1
2 4
8a3b2 4 a4 6x y Monomials
36. 
16a2b3  16b4
37. 43   38. 1
y 
  2 Monomials A monomial is a number, a variable, or the product of a number and one or
3x y x6 x y2 more variables. Constants are monomials that contain no variables.

12x3y2z8 2x3y2
1 1
an  
3 n  a for any real number a  0 and any integer n.
n
n and 

30a2b6 a4b2
Negative Exponent
1
39. 2

a a
v
3 6 ★ 40.   
60a6b8
★ 41.   
30x6y4z1 5z7
When you simplify an expression, you rewrite it without parentheses or negative
w vw 2 exponents. The following properties are useful when simplifying expressions.

Product of Powers am  an  am  n for any real number a and integers m and n.

Lesson 5-1
am

★ 42. If 2r  5  22r  1, what is the value of r? 6

Quotient of Powers   am  n for any real number a  0 and integers m and n.
an
For a, b real numbers and m, n integers:
(am )n  amn
(ab)m  ambm

★ 43. What value of r makes y28  y3r  y7 true? 7

Properties of Powers
 ab   
n an
,b0
bn
n
 ab    ab  or 
n bn
, a  0, b  0
an

Express each number in scientific notation. Example Simplify. Assume that no variable equals 0.

44. 462.3 4.623  45. 43,200 4.32  46. 0.0001843 1.843  104
(m4)3
102 104 a. (3m4n2)(5mn)2
(3m4n2)(5mn)2  3m4n2  25m2n2
b. 
2 2(2m )
(m4)3 m12
 75m4m2n2n2  
47. 0.006810 6.81  103 48. 502,020,000 49. 675,400,000 6.754  108
(2m2)2 1
 75m4  2n2  2  4 4m
 75m6  m12  4m4
5.0202  108  4m16

Evaluate. Express the result in scientific notation. Exercises

50. (4.15
103)(3.0
106) 1.245  1010 51. (3.01
102)(2
103) 6.02  105 Simplify. Assume that no variable equals 0.
b8
1. c12  c4  c6 c14 2. 2 b 6 3. (a4)5 a 20
b
6.3
10 5 9.3
10 7
52. 3 4.5  102 53.  3 6.2  10
10
1.4
10 1.5
10 x2 y
4 1 
4. 
y2
5.  aa bb 
3 2
2 1 b
  xxyy 
6. 3
2 2 x2

6 a5 y4
55. (4.1
104)2 1.681  107
x y x
54. (6.5
104)2 4.225  109
1 8m3n2 2m2
7.  (5a2b3)2(abc)2 5a6b 8c 2 8. m7  m8 m15 9. 3 
5 4mn n
56. POPULATION The population of Earth is about 6,080,000,000. Write this
number in scientific notation. 6.08  109 10. 
23c4t2
2 4 2 2
2 c t
11. 4j(2j2k2)(3j 3k7) 
5
24j 2
k
12. 
2mn2(3m2n)2 3
3 4 m 2
12m n 2

57. BIOLOGY Use the diagram at the right Gl NAME

/M G ______________________________________________
Hill 239 DATE ____________
GlPERIOD
Al _____
b 2

to write the diameter of a typical flu virus Skills

5-1 Practice,
Practice (Average)
p. 241 and
in scientific notation. 2  107 m Practice,
Monomialsp. 242 (shown)
Simplify. Assume that no variable equals 0.

1. n5  n2 n7 2. y7  y3  y2 y12
58. CHEMISTRY One gram of water contains 1
about 3.34
1022 molecules. About how 3. t9  t8 t 4. x4  x4  x4 4 x
8c9
0.0000002 m 5. (2f 4)6 64f 24 6. (2b2c3)3  
many molecules are in 500 grams of water? 6 b

1.67  1025
20d 3t 2
7. (4d 2t5v4)(5dt3v1)  5 8. 8u(2z)3 64uz 3
v
59. RESEARCH Use the Internet or other source 9. 
12m8 y6
4 
9my
4m 7y 2
3
10. 
6s5x3
7 
18sx4
s4
3x
to find the masses of Earth and the Sun. About 27x3(x7) 27x 6
11. 
16x4

16
12. 
2 3 6
2
3r s z
2

4
9r 4s 6z 12
how many times as large as Earth is the Sun? 13. (4w3z5)(8w)2  5
256
wz
14. (m4n6)4(m3n2p5)6 m 34n 36p 30

about 330,000 times  32

15.  d 2f 4  43 d 5f
4 3
12d 23f 19 16.  x
2 5 y 
2x y 3 2 2

y6
4x 2
60. CRITICAL THINKING Determine which is 17. 
(3x2y3)(5xy8) 15x11
3 4 2 
20(m2v)(v)3
18.  
4v2

greater, 10010 or 10100. Explain.

3
(x ) y y 2 4 2
5(v) (m ) m

10010  (102)10 or 1020, and 10100 1020, so 10100 10010.

Express each number in scientific notation.

19. 896,000 20. 0.000056 21. 433.7

108
CRITICAL THINKING For Exercises 61 and 62, use the following proof of the 8.96  105 5.6  105 4.337  1010

Power of a Power Property. 61. Definition of an exponent Evaluate. Express the result in scientific notation.
2.7
106
22. (4.8
102)(6.9
104) 23. (3.7
109)(8.7
102) 24. 
9
10
10
m factors n factors 3.312  107 3.219  1012 3  105



A scatter plot of aman  a  a  …  a  a  a  …  a 25. COMPUTING The term bit, short for binary digit, was first used in 1946 by John Tukey.
A single bit holds a zero or a one. Some computers use 32-bit numbers, or strings of
populations will help m  n factors 32 consecutive bits, to identify each address in their memories. Each 32-bit number
corresponds to a number in our base-ten system. The largest 32-bit number is nearly
4,295,000,000. Write this number in scientific notation. 4.295  109
you make a model for aa…a 26. LIGHT When light passes through water, its velocity is reduced by 25%. If the speed of
the data. Visit  am  n light in a vacuum is 1.86
105 miles per second, at what velocity does it travel through
water? Write your answer in scientific notation. 1.395  105 mi/s
www.algebra2.com/ 27. TREES Deciduous and coniferous trees are hard to distinguish in a black-and-white

webquest to continue 61. What definition or property allows you to make each step of the proof? photo. But because deciduous trees reflect infrared energy better than coniferous trees,
the two types of trees are more distinguishable in an infrared photo. If an infrared

work on your 62. Prove the Power of a Product Property, (ab)m  ambm. See margin. wavelength measures about 8
107 meters and a blue wavelength measures about
4.5
107 meters, about how many times longer is the infrared wavelength than the
blue wavelength? about 1.8 times
WebQuest project. Gl NAME
/M G ______________________________________________
Hill 242 DATE ____________
Gl PERIOD
Al _____
b 2

63. WRITING IN MATH Answer the question that was posed at the beginning Reading
5-1 Readingto
to Learn
Learn Mathematics
of the lesson. See margin. Mathematics, p. 243 ELL
Monomials
Pre-Activity Why is scientific notation useful in economics?
Why is scientific notation useful in economics? Read the introduction to Lesson 5-1 at the top of page 222 in your textbook.
Your textbook gives the U.S. public debt as an example from economics that
Include the following in your answer: involves large numbers that are difficult to work with when written in
standard notation. Give an example from science that involves very large
• the 2000 national debt of $5,674,200,000,000 and the U.S. population of numbers and one that involves very small numbers. Sample answer:
distances between Earth and the stars, sizes of molecules
and atoms
281,000,000, both written in words and in scientific notation, and

Lesson 5-1
• an explanation of how to find the amount of debt per person, with the
Reading the Lesson
result written in scientific notation and in standard notation. 1. Tell whether each expression is a monomial or not a monomial. If it is a monomial, tell
whether it is a constant or not a constant.

www.algebra2.com/self_check_quiz Lesson 5-1 Monomials 227 a. 3x2 monomial; not a constant b. y2  5y  6 not a monomial

1
c. 73 monomial; constant d. 
z not a monomial

2. Complete the following definitions of a negative exponent and a zero exponent.

NAME ______________________________________________ DATE ____________ PERIOD _____ 1

For any real number a  0 and any integer n, an  an .
Enrichment,
5-1 Enrichment p. 244 For any real number a  0, a0  1 .

Properties of Exponents 3. Name the property or properties of exponents that you would use to simplify each
expression. (Do not actually simplify.)
The rules about powers and exponents are usually given with letters such as m, n,
and k to represent exponents. For example, one rule states that am  an  am  n. m8
a. 3 quotient of powers
m
In practice, such exponents are handled as algebraic expressions and the rules of
algebra apply. b. y6  y9 product of powers

Example 1 Simplify 2a2(a n  1  a 4n). c. (3r2s)4 power of a product and power of a power
2a2(an  1  a4n)  2a2  an  1  2a2  a4n Use the Distributive Law.

 2a2  n  1  2a2  4n Recall am  an  am  n.

Helping You Remember
 2an  3  2a2  4n Simplify the exponent 2  n  1 as n  3.

It is important always to collect like terms only. 4. When writing a number in scientific notation, some students have trouble remembering
when to use positive exponents and when to use negative ones. What is an easy way to
remember this? Sample answer: Use a positive exponent if the number is
Example 2 Simplify (a n  bm)2. 10 or greater. Use a negative number if the number is less than 1.
(an  bm)2  (an  bm)(an  bm)
F O I L
 an  an  an  bm  an  bm  bm  bm The second and third terms are like terms.

Lesson 5-1 Monomials 227

 (2x2)3
Standardized 64. Simplify 
4 . D

4 Assess Test Practice A

x

2
12x
B
2x

3
C
1

2x2
D
2x2

3
65. 7.3
105  ? B
Open-Ended Assessment A 73,000 B 730,000 C 7,300,000 D 73,000,000
Speaking Have students explain
in informal language how to tell
whether a number is written in
Maintain Your Skills
scientific notation. Then have
them explain how to simplify Mixed Review Solve each system of equations by using inverse matrices. (Lesson 4-8)
monomial expressions involving 66. 2x  3y  8 (1, 2) 67. x  4y  9 (3, 3)
negative exponents. x  2y  3 3x  2y  3

Find the inverse of each matrix, if it exists. (Lesson 4-7)


2 5 1 3

Intervention 68.  12 5
2  1 2 69. 42 31 2

2
1 2
New Simplifying
expressions Evaluate each determinant. (Lesson 4-3)
3

1
with exponents
is a skill that is
70.
32 0
2
6
71. 2
3
1
0
4 7
2
needed frequently in algebra.
Solve each system of equations. (Lesson 3-5)
Take time to clear up any mis-
72. x  y  5 (2, 3, 1) 73. a  b  c  6 (2, 0, 4)
conceptions at this point and xyz4 2a  b  3c  16
to help students develop an 2x  y  2z  1 a  3b  2c  6
understanding of the properties
so that they remember the pro- TRANSPORTATION For Exercises
cedures correctly for later use. 74–76, refer to the graph at the USA TODAY Snapshots®
right. (Lesson 2-5)
74. See margin. 74. Make a scatter plot of the data, Hanging on to the old buggy
The median age of automobiles and trucks on
where the horizontal axis is the the road in the USA:
Getting Ready for number of years since 1970.
1970 4.9 years
Lesson 5-2 75. Sample answer 75. Write a prediction equation.
using (0, 4.9) and 76. Predict the median age of 1975 5.4 years
PREREQUISITE SKILL Lesson 5-2 (28, 8.3): y  vehicles on the road in 2010.
presents multiplying polynomials. 0.12x  4.9 Sample answer: 9.7 yr
1980 6 years
This multiplication involves the 1985 6.9 years
use of the Distributive Property.
1990 6.5 years
Exercises 79–84 should be used
to determine your students’ 1995 7.7 years
familiarity with the Distributive 1999 8.3 years
Property. Source: Transportation Department

By Keith Simmons, USA TODAY

Answer Solve each equation. (Lesson 1-3)

74. Median Age of Vehicles 77. 2x  11  25 7 78. 12  5x  3 3
y
Getting Ready for Use the Distributive Property to find each product.
Median Age (yr)

8
7
the Next Lesson (To review the Distributive Property, see Lesson 1-2.)
79. 2(x  y) 2x  2y 80. 3(x  z) 3x  3z 81. 4(x  2) 4x  8
6
82. 2(3x  5) 6x  10 83. 5(x  2y) 5x  10y 84. 3(y  5) 3y  15
5
228 Chapter 5 Polynomials
4
0
0 10 20 30 x
Years Since 1970 Online Lesson Plans
USA TODAY Education’s Online site offers resources and
interactive features connected to each day’s newspaper.
Experience TODAY, USA TODAY’s daily lesson plan, is
available on the site and delivered daily to subscribers.
This plan provides instruction for integrating USA TODAY
graphics and key editorial features into your mathematics
classroom. Log on to www.education.usatoday.com.

228 Chapter 5 Polynomials

 Polynomials Lesson
Notes

• Add and subtract polynomials. Year Tuition

• Multiply polynomials. 1
2
$13,872
$14,427
1 Focus
Vocabulary can polynomials be applied 3 $15,004

• polynomial to financial situations? 4 $15,604

5-Minute Check
• terms Shenequa wants to attend Purdue University in Transparency 5-2 Use as a
• like terms Indiana, where the out-of-state tuition is $13,872. quiz or review of Lesson 5-1.
• trinomial Suppose the tuition increases at a rate of 4% per
• binomial year. You can use polynomials to represent the Mathematical Background notes
• FOIL method increasing tuition costs. are available for this lesson on
p. 220C.
ADD AND SUBTRACT POLYNOMIALS If r represents the rate of increase can polynomials be
of tuition, then the tuition for the second year will be 13,872(1  r). For the third applied to financial
year, it will be 13,872(1  r)2, or 13,872r2  27,744r  13,872 in expanded form. The
expression 13,872r2  27,744r  13,872 is called a polynomial. A polynomial is a situations?
monomial or a sum of monomials. Ask students:
Study Tip The monomials that make up a polynomial are called the terms of the polynomial.
In a polynomial such as x2  2x  x  1, the two monomials 2x and x can be
• What is meant by “tuition
Reading Math combined because they are like terms. The result is x2  3x  1. The polynomial increases at a rate of 4% per
The prefix bi- means x2  3x  1 is a trinomial because it has three unlike terms. A polynomial such as year?” Each year the tuition is 4%
two, and the prefix xy  z3 is a binomial because it has two unlike terms. The degree of a polynomial higher than it was the year before.
tri- means three. is the degree of the monomial with the greatest degree. For example, the degree of
x2  3x  1 is 2, and the degree of xy  z3 is 3. • Will the amount of the tuition
increase be the same each
Example 1 Degree of a Polynomial year? no
Determine whether each expression is a polynomial. If it is a polynomial, state
the degree of the polynomial.
1
a. x3y5  9x4
6
2 Teach
This expression is a polynomial because each term is a monomial.
ADD AND SUBTRACT
The degree of the first term is 3  5 or 8, and the degree of the second term is 4.
The degree of the polynomial is 8. POLYNOMIALS
b. x  x  5 In-Class Examples Power
Point®
This expression is not a polynomial because x is not a monomial.
1 Determine whether each
To simplify a polynomial means to perform the operations indicated and combine expression is a polynomial. If
like terms. it is a polynomial, state the
degree of the polynomial.
Example 2 Subtract and Simplify
Simplify (3x2  2x  3)  (x2  4x  2).
a. c4  4c  18 no
(3x2  2x  3)  (x2  4x  2)  3x2  2x  3  x2  4x  2 3
Distribute the 1. b. 16p5   p2 q7 yes, 9
4
 (3x2  x2)  (2x  4x)  (3  2) Group like terms.

 2x2  6x  5 Combine like terms. 2 Simplify (2a3  5a  7) 

(a3  3a  2). a3  8a  9
www.algebra2.com/extra_examples Lesson 5-2 Polynomials 229

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 5 Resource Masters School-to-Career Masters, p. 9 5-Minute Check Transparency 5-2
• Study Guide and Intervention, pp. 245–246 Teaching Algebra With Manipulatives Answer Key Transparencies
• Skills Practice, p. 247 Masters, p. 234
• Practice, p. 248 Technology
• Reading to Learn Mathematics, p. 249 Interactive Chalkboard
• Enrichment, p. 250

Lesson x-x Lesson Title 229

MULTIPLY POLYNOMIALS MULTIPLY POLYNOMIALS You can use the Distributive Property to multiply
polynomials.
In-Class Examples Power
Point® Example 3 Multiply and Simplify
3 Find y(4y2  2y  3). Find 2x(7x2  3x  5).

4y3  2y2  3y 2x(7x2  3x  5)  2x(7x2)  2x(3x)  2x(5) Distributive Property

 14x3  6x2  10x Multiply the monomials.
4 Find (2p  3)(4p  1).
8p2  14p  3 You can use algebra tiles to model the product of two binomials.
5 Find (a2  3a  4)(a  2).
a 3  5a2  2a  8
Multiplying Binomials
Use algebra tiles to find the product of x  5 and x  2.
3 Practice/Apply • Draw a 90° angle on your paper.
• Use an x tile and a 1 tile to mark off a
x 5

length equal to x  5 along the top. 2

x x x x x x x
• Use the tiles to mark off a length equal
Study Notebook to x  2 along the side.
x 1 1 1 1 1
• Draw lines to show the grid formed. 2
x 1 1 1 1 1
Have students— • Fill in the lines with the appropriate tiles
• add the definitions/examples of to show the area product. The model
the vocabulary terms to their shows the polynomial x2  7x  10.
The area of the rectangle is the product of its length and width. Substituting
Vocabulary Builder worksheets for for the length, width, and area with the corresponding polynomials, we find
Chapter 5. that (x  5)(x  2)  x2  7x  10.
• include any other item(s) that they
find helpful in mastering the skills In Example 4, the FOIL method is used to multiply binomials. The FOIL method
in this lesson. is an application of the Distributive Property that makes the multiplication easier.

FOIL Method for Multiplying Binomials

The product of two binomials is the sum of the products of F the first terms,
About the Exercises… O the outer terms, I the inner terms, and L the last terms.
Organization by Objective
• Add and Subtract
Study Tip Example 4 Multiply Two Binomials
Polynomials: 16–27
• Multiply Polynomials: Vertical Method Find (3y  2)(5y  4).
You may also want to use
28–33, 37–50 (3y  2)(5y  4)  3y  5y  3y  4  2  5y  24
the vertical method to




multiply polynomials. First terms Outer terms Inner terms Last terms
Odd/Even Assignments 3y  2  15y2  22y  8 Multiply monomials and add like terms.
Exercises 16–33 and 37–50 are (
) 5y  4
12y  8
structured so that students 15y2  10y
practice the same concepts 15y2  22y  8 Example 5 Multiply Polynomials
whether they are assigned Find (n2  6n  2)(n  4).
odd or even problems. (n2  6n  2)(n  4)
 n2(n  4)  6n(n  4)  (2)(n  4) Distributive Property
Assignment Guide  n2  n  n2  4  6n  n  6n  4  (2)  n  (2)  4 Distributive Property
Basic: 17–33 odd, 35, 36,  n3  4n2  6n2  24n  2n  8 Multiply monomials.
37–45 odd, 51, 53–69  n3  10n2  22n  8 Combine like terms.
Average: 17–33 odd, 35, 36,
37–53 odd, 54–69 230 Chapter 5 Polynomials

Advanced: 16–34 even, 35, 36,

38–52 even, 54–65 (optional:
66–69)
Answers Algebra Activity
22. 4x 2  3x  7 Materials: protractor, ruler/straightedge, algebra tiles
23. 3y  3y 2 • Remind students that the length of an x tile is not
a multiple of the length of a side of a unit tile.
24. r 2  r  6
• Point out to students that the width of an x tile is
25. 10m 2  5m  15 exactly one unit (the same as the length of a side
26. 4x 2  3xy  6y 2 of a unit tile).

230 Chapter 5 Polynomials

 NAME ______________________________________________ DATE ____________ PERIOD _____

Study
5-2 Guide
Study andIntervention
Guide and Intervention,
p. 245 (shown) and p. 246
Polynomials
Concept Check 1. OPEN ENDED Write a polynomial of degree 5 that has three terms. Add and Subtract Polynomials
Polynomial a monomial or a sum of monomials
1. Sample answer: 2. Identify the degree of the polynomial 2x3  x2  3x4  7. 4 Like Terms terms that have the same variable(s) raised to the same power(s)

x5  x4  x3 3. Model 3x(x  2) using algebra tiles. See pp. 283A–283B.

To add or subtract polynomials, perform the indicated operations and combine like terms.

Example 1 Simplify 6rs  18r 2  5s2  14r 2  8rs  6s2.

6rs  18r2  5s2 14r2  8rs 6s2
 (18r2  14r2)  (6rs  8rs)  (5s2  6s2) Group like terms.

Guided Practice Determine whether each expression is a polynomial. If it is a polynomial, state  4r2  2rs  11s2

Example 2
Combine like terms.

the degree of the polynomial. 4xy2 12xy 

Simplify 4xy2  12xy  7x 2y  (20xy  5xy2  8x 2y).
 (20xy 
7x2y  5xy2 8x2y)
1 mw  3 2  4xy2  12xy  7x2y  20xy  5xy2  8x2y
4. 2a  5b yes, 1 5. x3  9y yes, 3 6.  3  no
Distribute the minus sign.
 (7x2y  8x2y )  (4xy2  5xy2)  (12xy  20xy)
GUIDED PRACTICE KEY 3 nz  1
Group like terms.

Lesson 5-2
 x2y  xy2  8xy Combine like terms.

Exercises Examples
Simplify. 8. 3x2  7x  8 10. 10p3q2  6p5q3  8p3q5 Exercises
Simplify.
4–6 1
7. (2a  3b)  (8a  5b) 10a  2b 8. (x2  4x  3)  (4x2  3x  5) 1. (6x2  3x  2)  (4x2  x  3) 2. (7y2  12xy  5x2)  (6xy  4y2  3x2)
7, 8 2 2x 2  4x  5 3y 2 18xy  8x 2
9, 10 3 9. 2x(3y  9) 6xy  18x 10. 2p2q(5pq  3p3q2  4pq4) 3. (4m2  6m)  (6m  4m2) 4. 27x2  5y2  12y2  14x2
8m 2  12m 13x 2  7y 2
11–14 4 11. (y  10)(y  7) y2  3y  70 12. (x  6)(x  3) x2  9x  18 5. (18p2  11pq  6q2)  (15p2  3pq  4q2) 6. 17j 2  12k2  3j 2  15j 2  14k2
15 5 3p 2  14pq  10q 2 5j 2  2k 2
13. (2z  1)(2z  1) 4z2  1 14. (2m  3n)2 4m2  12mn  9n2 7. (8m2  7n2)  (n2  12m2) 8. 14bc  6b  4c  8b  8c  8bc
20m 2  8n 2 14b  22bc  12c
9. 6r2s  11rs2  3r2s  7rs2  15r2s  9rs2 10. 9xy  11x2  14y2  (6y2  5xy  3x2)

Application 15. GEOMETRY Find the area of the triangle. 24r 2s  5rs 2 14x 2  4xy  20y 2

7.5x2  12.5x ft2 11. (12xy  8x  3y)  (15x  7y  8xy)

7x  4xy  4y
12. 10.8b2  5.7b  7.2  (2.9b2  4.6b  3.1)
7.9b 2  1.1b  10.3
13. (3bc  9b2  6c2)  (4c2  b2  5bc) 14. 11x2  4y2  6xy  3y2  5xy  10x2
5x ft 10b 2  8bc  2c 2 x 2  xy  7y 2
1 3 1 1 1 3
15.  x2   xy   y2   xy   y2   x2 16. 24p3  15p2  3p  15p3  13p2  7p
4 8 2 2 4 8
1 7 3
  x 2   xy   y 2 9p 3  2p 2  4p
8 8 4

3x  5 ft Gl NAME
/M G ______________________________________________
Hill 245 DATE ____________
GlPERIOD
Al _____
b 2

★ indicates increased difficulty Skills

5-2 Practice,
Practice (Average)
p. 247 and
Practice, p. 248 (shown)
Polynomials
Practice and Apply Determine whether each expression is a polynomial. If it is a polynomial, state the
degree of the polynomial.
4 12m8n9
1. 5x3  2xy4  6xy yes; 5 2.   ac  a5d3 yes; 8 3. 2 no
3 (m  n)
Homework Help Determine whether each expression is a polynomial. If it is a polynomial, state 4. 25x3z  x78
 yes; 4 5. 6c2  c  1 no
5 6
6.   
r s
no
For See the degree of the polynomial. Simplify.
Exercises Examples 6xy 3c
16–21 1
16. 3z2  5z  11 yes, 2 17. x3  9 yes, 3 18.    no 7. (3n2  1) 
11n 2  7
(8n2  8) 8. (6w  11w2)
18w 2  6w  4
 (4  7w2)

z d
9. (6n  13n2)  (3n  9n2) 10. (8x2  3x)  (4x2  5x  3)
22–27, 35, 2 4 5 9n  4n 2 4x 2  8x  3
36, 51 19. 
m  5 no 20. 5x2y4  x3 yes, 6 21. y2  y7 yes, 7 11. (5m2  2mp  6p2)  (3m2  5mp  p2) 12. (2x2  xy  y2)  (3x2  4xy  3y2)
28–33, 3
3 6 8m 2  7mp  7p 2 x 2  3xy  4y 2
47, 48 13. (5t  7)  (2t2  3t  12) 14. (u  4)  (6  3u2  4u)

34 2, 3
Simplify. 22–33. See margin. 2t 2  8t  5 3u 2  5u  10
15. 9( y2  7w) 16. 9r4y2(3ry7  2r3y4  8r10)
37–46, 4 22. (3x2  x  2)  (x2  4x  9) 23. (5y  3y2)  (8y  6y2) 9y 2  63w 27r 5y 9  18r 7y 6  72r14y 2
52, 53 17. 6a2w(a3w  aw4) 18. 5a2w3(a2w6  3a4w2  9aw6)

49, 50, 54 5 24. (9r2  6r  16)  (8r2  7r  10) 25. (7m2  5m  9)  (3m2  6) 6a 5w 2  6a 3w 5 5a4w 9  15a 6w 5  45a 3w 9
3
19. 2x2(x2  xy  2y2) 20.   ab3d2(5ab2d5  5ab)
26. (4x2  3y2  5xy)  (8xy  3y2) 27. (10x2  3xy  4y2)  (3x2  5xy) 2x 4  2x 3y  4x 2y 2
5
3a 2b 5d 7  3a 2b4d 2
Extra Practice 28. 4b(cb  zd) 29. 4a(3a2  b) 21. (v2  6)(v2  4) 22. (7a  9y)(2a  y)
v 4  2v 2  24 14a 2  11ay  9y 2
See page 837.
30. 5ab2(3a2b  6a3b  3a4b4) 31. 2xy(3xy3  4xy  2y4) 23. ( y  8)2
y 2  16y  64
24. (x2  5y)2
x 4  10x 2y  25y 2
25. (5x  4w)(5x  4w) 26. (2n4  3)(2n4  3)
3 1
32. x2(8x  12y  16xy2) 33. a3(4a  6b  8ab4) 25x 2  16w 2 4n8  9
4 2 27. (w  2s)(w2  2ws  4s2) 28. (x  y)(x2  3xy  2y2)
w3  8s3 x 3  2x 2y  xy 2  2y 3
34. PERSONAL FINANCE Toshiro wants to know how to invest the $850 he has 29. BANKING Terry invests $1500 in two mutual funds. The first year, one fund grows 3.8%
and the other grows 6%. Write a polynomial to represent the amount Terry’s $1500
saved. He can invest in a savings account that has an annual interest rate of grows to in that year if x represents the amount he invested in the fund with the lesser
growth rate. 0.022x  1590

3.7%, and he can invest in a money market account that pays about 5.5% per 30. GEOMETRY The area of the base of a rectangular box measures 2x2  4x  3 square
units. The height of the box measures x units. Find a polynomial expression for the
year. Write a polynomial to represent the amount of interest he will earn in volume of the box. 2x 3  4x 2  3x units3

1 year if he invests x dollars in the savings account and the rest in the money Gl
Reading
NAME
/M G ______________________________________________
Hill 248 DATE ____________
Gl PERIOD
Al _____
b 2

market account. 46.75  0.018x 5-2 Readingto

to Learn
Learn Mathematics
Mathematics, p. 249 ELL
Polynomials
Pre-Activity How can polynomials be applied to financial situations?
E-SALES For Exercises 35 and 36, use the following information. Read the introduction to Lesson 5-2 at the top of page 229 in your textbook.

A small online retailer estimates that the cost, in dollars, associated with selling Suppose that Shenequa decides to enroll in a five-year engineering program
rather than a four-year program. Using the model given in your textbook,

x units of a particular product is given by the expression 0.001x2  5x  500. how could she estimate the tuition for the fifth year of her program? (Do
not actually calculate, but describe the calculation that would be necessary.)

The revenue from selling x units is given by 10x. 35. 0.001x2  5x  500
Multiply $15,604 by 1.04.

35. Write a polynomial to represent the profit generated by the product. Reading the Lesson
1. State whether the terms in each of the following pairs are like terms or unlike terms.
36. Find the profit from sales of 1850 units. $5327.50 a. 3x2, 3y2 unlike terms b. m4, 5m4 like terms
c. 8r3, 8s3 unlike terms d. 6, 6 like terms

www.algebra2.com/self_check_quiz Lesson 5-2 Polynomials 231

Lesson 5-2
2. State whether each of the following expressions is a monomial, binomial, trinomial, or
not a polynomial. If the expression is a polynomial, give its degree.
a. 4r4  2r  1 trinomial; degree 4 b. 3x
 not a polynomial
c. 5x  4y binomial; degree 1 d. 2ab  4ab2  6ab3 trinomial; degree 4

NAME ______________________________________________ DATE ____________ PERIOD _____

3. a. What is the FOIL method used for in algebra? to multiply binomials

Answers Enrichment,
5-2 Enrichment p. 250
b. The FOIL method is an application of what property of real numbers?
Distributive Property
c. In the FOIL method, what do the letters F, O, I, and L mean?

27. 7x 2  8xy  4y 2
first, outer, inner, last
Polynomials with Fractional Coefficients d. Suppose you want to use the FOIL method to multiply (2x  3)(4x  1). Show the
terms you would multiply, but do not actually multiply them.

28. 4b c  4bdz
2 Polynomials may have fractional coefficients as long as there are no variables
in the denominators. Computing with fractional coefficients is performed in F (2x)(4x)
the same way as computing with whole-number coefficients. O (2x)(1)

29. 12a 3  4ab Simpliply. Write all coefficients as fractions.

I
L
(3)(4x)
(3)(1)

30. 15a 3b 3  30a 4b 3  15a 5b 6  35   73 

2 1 5 3 31 5 55
1. m  p  n  p  m  n m  n  p
7 3 2 4 10 12 21
Helping You Remember
31. 6x 2y 4  8x 2y 2  4xy 5  32 4 5
  1 2
  7 6
2.  x   y   z   x  y   z   x   y   z x  y  z
1

3 25 7
4. You can remember the difference between monomials, binomials, and trinomials by
thinking of common English words that begin with the same prefixes. Give two words
3 4 4 5 8 7 2 8 21 20 unrelated to mathematics that start with mono-, two that begin with bi-, and two that
32. 6x 3  9x 2y  12x 3y 2 begin with tri-. Sample answer: monotonous, monogram; bicycle, bifocal;
tricycle, tripod

33. 2a 4  3a 3b  4a 4b 4  12   56 
1 1 2 3 4 1 1
3.  a2   ab   b2   a2   ab   b2 a2  ab  b2
3 4 3 4 3 3 2

Lesson 5-2 Polynomials 231

 Simplify.

4 Assess 1
37. (p  6)(p  4) p2  2p  24
39. (b  5)(b  5)  25
b2 40.
38. (a  6)(a  3) a2  9a  18
(6  z)(6  z) 36  z2
48. xy3  y  
Open-Ended Assessment x 41. (3x  8)(2x  6) 6x2  34x  48 42. (4y  6)(2y  7) 8y2  16y  42
51. 9c2  12cd  7d 2 43. (a3  b)(a3  b) a6  b2 44. (m2  5)(2m2  3) 2m4  7m2  15
Writing Have students write an
45. (x  3y) x  6xy  9y
2 2 2 46. (1  4c)2 1  8c  16c2
explanation, including an 1
example, showing why the FOIL ★ 47. d3(d5  2d3  d1) d 2  2  4 ★ 48. x3y2(yx4  y1x3  y2x2)
R W d
method is a valid alternative to ★ 49. (3b  c)327b3  27b2c  9bc2  c3 ★ 50. (x2  xy  y2)(x  y) x3  y3
applying the Distributive R 51. Simplify (c2  6cd  2d2)  (7c2  cd  8d2)  (c2  5cd  d2).
Property when multiplying two RR RW 52. Find the product of 6x  5 and 3x  2. 18x2  27x  10
binomials.
W 53. GENETICS Suppose R and W represent two genes that a plant can inherit
RW WW from its parents. The terms of the expansion of (R  W)2 represent the possible
Getting Ready for pairings of the genes in the offspring. Write (R  W)2 as a polynomial.
Lesson 5-3 Genetics R2  2RW  W 2
The possible genes of 54. CRITICAL THINKING What is the degree of the product of a polynomial of
PREREQUISITE SKILL Lesson 5-3 parents and offspring degree 8 and a polynomial of degree 6? Include an example in support of your
presents dividing polynomials. can be summarized in a answer. 14; Sample answer: (x8  1)(x6  1)  x14  x8  x6  1
Punnett square, such as
Dividing polynomials requires the one above.
55. WRITING IN MATH Answer the question that was posed at the beginning of
the use of the properties of Source: Biology: The Dynamics
the lesson. See pp. 283A–283B.
of Life
exponents. Exercises 66–69 How can polynomials be applied to financial situations?
should be used to determine
Include the following in your answer:
your students’ familiarity with
• an explanation of how a polynomial can be applied to a situation with a fixed
the properties of exponents. percent rate of increase,
• two expressions in terms of r for the tuition in the fourth year, and
• an explanation of how to use one of the expressions and the 4% rate of
Answers increase to estimate Shenequa’s tuition in the fourth year, and a comparison
of the value you found to the value given in the table.
63. y
Standardized 56. Which polynomial has degree 3? D
Test Practice A x3  x2  2x4 B 2x2  3x  4
C x2  x  123 D 1  x  x3
O x 57. (x  y)  (y  z)  (x  z)  ? B
y   13 x  2 A 2x  2y  2z B 2z
C 2y D xyz

Maintain Your Skills

64. y
Mixed Review Simplify. Assume that no variable equals 0. (Lesson 5-1)
x2yz4 xz2 3ab2 2 b2
61. 
6a2b  4a2
58. (4d2)3 64d 6 59. 5rt2(2rt)2 20r 3t 4
60. 32  
x  y 2 xy z y2
62. Solve the system 4x  y  0, 2x  3y  14 by using inverse matrices.
O x (Lesson 4-8) (1, 4)
Graph each inequality. (Lesson 2-7) 63–65. See margin.
1
63. y  x  2 64. x  y 2 65. 2x  y  1
3
Getting Ready for PREREQUISITE SKILL Simplify. Assume that no variable equals 0.
65. y the Next Lesson (To review properties of exponents, see Lesson 5-1.)
x3 4y5 x2y3 3
9a b
66.  x2 67. 2 2y3 68.  xy2 69.  3a2
x 2y xy 3ab
232 Chapter 5 Polynomials
2x  y  1
O x

Differentiated Instruction
Logical Have students demonstrate how to use algebra tiles to multiply
two binomials that contain at least one negative coefficient.

232 Chapter 5 Polynomials

 Dividing Polynomials Lesson
Notes

• Divide polynomials using long division.

• Divide polynomials using synthetic division.
1 Focus
Vocabulary can you use division of polynomials in manufacturing?
• synthetic division A machinist needed 32x2  x square inches of metal to make a square pipe 5-Minute Check
8x inches long. In figuring the area needed, she allowed a fixed amount of metal Transparency 5-3 Use as a
for overlap of the seam. If the width of the finished pipe will be x inches, how quiz or review of Lesson 5-2.
wide is the seam? You can use a quotient of polynomials to help find the answer.
Metal Needed Finished Pipe
Mathematical Background notes
s
are available for this lesson on
x
2
p. 220C.
x
x
x
8x can you use division of
x
x polynomials in
x
2
8x
manufacturing?
s  width of seam Ask students:
x
• What does the expression 
2
USE LONG DIVISION In Lesson 5-1, you learned to divide monomials. You shown in the figure represent?
can divide a polynomial by a monomial by using those same skills.
one half of the side length of the
Example 1 Divide a Polynomial by a Monomial pipe opening
4x3y2  8xy2  12x2y3 • What happens to the width of
Simplify  .
4xy the pipe opening as the length of
4x3y2  8xy2  12x2y3 4x3y2 8xy2 12x2y3 the pipe increases? The width of
       Sum of quotients
4xy 4xy 4xy 4xy the pipe opening, x, increases also.
4 8 12
   x3  1y2  1    x1  1y2  1    x2  1y3  1 Divide.
4 4 4
 x2y  2y  3xy2 x1  1  x0 or 1

2 Teach
You can use a process similar to long division to divide a polynomial by a
polynomial with more than one term. The process is known as the division algorithm. USE LONG DIVISION
When doing the division, remember that you can only add or subtract like terms.
In-Class Example Power
Point®
Example 2 Division Algorithm
5a b  15ab  10a b
2 3 3 4
Use long division to find (z2  2z  24)
(z  4). 1 Simplify  5ab
.
z z 6
z  4z2
z
24
2 z  4z2z
24
2 a  3b 2  2a2b 3
()z2  4z z(z  4)  z2  4z ()z2  4z
6z  24 2z  (4z)  6z 6z  24
()6z  24
0
The quotient is z  6. The remainder is 0.

Lesson 5-3 Dividing Polynomials 233

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 5 Resource Masters 5-Minute Check Transparency 5-3
• Study Guide and Intervention, pp. 251–252 Answer Key Transparencies
• Skills Practice, p. 253
• Practice, p. 254 Technology
• Reading to Learn Mathematics, p. 255 Interactive Chalkboard
• Enrichment, p. 256
• Assessment, p. 307

Lesson x-x Lesson Title 233

 Just as with the division of whole numbers, the division of two polynomials may
In-Class Examples Power
Point® result in a quotient with a remainder. Remember that 9  4  2  R1 and is often
1
written as 2. The result of a division of polynomials with a remainder can be
4
2 Use long division to find written in a similar manner.
(x2  2x  15)  (x  5).
x3
Standardized Example 3 Quotient with Remainder
Teaching Tip If students are Test Practice Multiple-Choice Test Item
having difficulty with the use of
the division algorithm, review
Which expression is equal to (t2  3t  9)(5  t)1?
the algorithm as it is used for
31
long division of numbers. A t8 
5t
B t  8
31 31
3 Which expression is equal to C t  8   
5t
D t  8   
5t
(a2  5a  3)(2  a)1? D
Read the Test Item
Aa3 Since the second factor has an exponent of 1, this is a division problem.
3 t2  3t  9
B a  3   (t2  3t  9)(5  t)1  
2a 5t
3
C a  3   Solve the Test Item
2a
t  8
3
D a  3   t  5t2t
39
 For ease in dividing, rewrite 5  t as t  5.
2a ()t2  5t t(t  5)  t2  5t
Test-Taking Tip
You may be able to 8t  9 3t  (5t)  8t
eliminate some of the ()8t  40 8(t  5)  8t  40
answer choices by 31 Subtract. 9  (40)  31
USE SYNTHETIC DIVISION substituting the same
value for t in the original The quotient is t  8, and the remainder is 31. Therefore,
In-Class Example Power
Point®
expression and the answer 31
(t2  3t  9)(5  t)1  t  8  . The answer is C.
choices and evaluating. 5t
Teaching Tip When discussing
Example 4, stress that the divisor
must be of the form x  r in
order to use synthetic division. USE SYNTHETIC DIVISION Synthetic 5x2  3x  4
division is a simpler process for dividing a x
x  253
3
1x
2
0
1x
8
4 Use synthetic division to find polynomial by a binomial. Suppose you want ()5x3  10x2
(x3  4x2  6x  4)  (x  2). to divide 5x3  13x2  10x  8 by x  2 using 3x2  10x
x2  2x  2 long division. Compare the coefficients in this ()3x2  6x
division with those in Example 4. 4x  8
()4x  8
Teaching Tip Ask students to dis- 0
cuss whether they would rather use
long division or synthetic division,
giving a reason for their choice. Example 4 Synthetic Division
Use synthetic division to find (5x3  13x2  10x  8)
(x  2).

Step 1 Write the terms of the dividend so that the 5x3  13x2  10x  8
degrees of the terms are in descending order.
←

←
Then write just the coefficients as shown at 5 13 10 8
the right.
Step 2 Write the constant r of the divisor x  r 2 5 13 10 8
Standardized to the left. In this case, r  2. Bring the
Test Practice first coefficient, 5, down as shown. 5
234 Chapter 5 Polynomials

Example 3 Point out that the

denominator 5  t is rewritten as
t  5 before starting the division
in order to have both numerator and Teacher to Teacher
denominator written in descending
Christine Waddell Albion M.S., Sandy, UT
order of the variable. Point out that
the first step in the long division "To help students better understand the division algorithm for polynomials,
eliminates choice A. Students could I first work through a long division problem with large whole numbers, such
also quickly eliminate choice B by as 3248 ÷ 24, step by step. Then I work through Example 2 and point out the
multiplying t  8 by t  5 and
noting that the product is not
similarities in each process."
t2  3t  9.
234 Chapter 5 Polynomials
 Step 3 Multiply the first coefficient by r: 2  5  10. 2 5 13 10 8
Write the product under the second ←
10
In-Class Example Power
Point®
coefficient. Then add the product and the 5 3
second coefficient: 13  10  3. 5 Use synthetic division to find
Step 4 Multiply the sum, 3, by r: 2(3)  6. 2 5 13 10 8 (4y4  5y2  2y  4) 
Write the product under the next 10 6
← (2y  1).
coefficient and add: 10  (6)  4. 4
5 3 4 2y 3  y 2  2y  
2y  1
Step 5 Multiply the sum, 4, by r: 2  4  8. 2 5 13 10 8
Write the product under the next 10 6 ←8 Teaching Tip Remind students
coefficient and add: 8  8  0. to include a coefficient of 0 for
The remainder is 0. 5 3 4
any missing terms of the
The numbers along the bottom row are the coefficients of the quotient. Start with variable in the dividend.
the power of x that is one less than the degree of the dividend. Thus, the quotient
is 5x2  3x  4.

To use synthetic division, the divisor must be of the form x  r. If the coefficient
of x in a divisor is not 1, you can rewrite the division expression so that you can use
synthetic division.

Example 5 Divisor with First Coefficient Other than 1

Use synthetic division to find (8x4  4x2  x  4)
(2x  1).
Use division to rewrite the divisor so it has a first coefficient of 1.
8x4  4x2  x  4 (8x4  4x2  x  4)  2 Divide numerator and
  
2x  1 (2x  1)  2 denominator by 2.
1
4x4  2x2  x  2
2 Simplify the numerator
 
1 and denominator.
x  
2

Since the numerator does not have an x3-term, use a coefficient of 0 for x3.
1 1
x  r  x  , so r  .
2 2

1 1
 4 0 2  2
2 2
1 1
2 1  
2 2
4 2 1 1 3

2
3

2
The result is 4x3  2x2 x1 1
. Now simplify the fraction.
x  
2
3

2
  3  x  1
1
x   2 2   Rewrite as a division expression.
2
3 2x  1 1 2x 1 2x  1
    x        
2 2 2 2
2 2
3 2
    Multiply by the reciprocal.
2 2x  1
3
  Multiply.
2x  1
3
The solution is 4x3  2x2  x  1  .
2x  1
(continued on the next page)
www.algebra2.com/extra_examples Lesson 5-3 Dividing Polynomials 235

Unlocking Misconceptions
• Subtracting Have students analyze any errors they make when using
long division. Verify that they are using the signs correctly.
• Remainders Have students do a simple numeric division example,
such as 8  5, to help them remember how to write the remainder
as part of the quotient.

Lesson 5-3 Dividing Polynomials 235

0220-283B Alg 2 Ch05-828000 11/22/02 7:02 PM Page 236

CHECK Divide using long division.

3 Practice/Apply 4x3 ⫺ 2x2 ⫺ x ⫹ 1
苶x苶4苶⫹
2x ⫹ 1冄8 苶苶x
0苶3苶⫺
苶苶x
4苶2苶⫹
苶苶x苶⫹苶4
(⫺)8x4 ⫹ 4x3
Study Notebook ⫺4x3 ⫺ 4x2
(⫺)⫺4x3 ⫺ 2x2
Have students— ⫺2x2 ⫹ x
• add the definitions/examples of (⫺)⫺2x2 ⫺ x
the vocabulary terms to their 2x ⫹ 4
Vocabulary Builder worksheets for (⫺)2x ⫹ 1
3
Chapter 5. 3
The result is 4x3 ⫺ 2x2 ⫺ x ⫹ 1 ⫹ ᎏᎏ. ⻫
• add the Test-Taking Tip on p. 234 2x ⫹ 1
to their list of tips which they can
review as they prepare for
standardized tests.
• include any other item(s) that they Concept Check 1. OPEN ENDED Write a quotient of two polynomials such that the remainder
is 5. Sample answer: (x2  x  5)  (x  1)
find helpful in mastering the skills 2. The divisor contains x ⫺ 3x ⫹ 1
2. Explain why synthetic division cannot be used to simplify ᎏ ᎏ.
3

in this lesson. an x 2 term. x2 ⫹ 1

3. FIND THE ERROR Shelly and Jorge are dividing x3 ⫺ 2x2 ⫹ x ⫺ 3 by x ⫺ 4.
3. Jorge; Shelly is
subtracting in the Shelly Jorge
columns instead of 4 1 -2 1 -3 4 1 -2 1 -3
adding. 4 -24 100 4 8 36
About the Exercises… 10. x 2  11x
34  1 -6 25 -103 1 2 9 33
60
Organization by Objective ᎏᎏ
x2 Who is correct? Explain your reasoning.
• Use Long Division: 15–20,
11. b3  b
1
45–48
Guided Practice Simplify. 7. 3a3
9a2  7a
6 8. z4  2z3  4z 2  5z  10
• Use Synthetic Division:
6xy2 ⫺ 3xy ⫹ 2x2y
21–44, 49, 50 GUIDED PRACTICE KEY 4. ᎏᎏᎏ 6y
3  2x 5. (5ab2 ⫺ 4ab ⫹ 7a2b)(ab)⫺1 5b
4  7a
xy
Odd/Even Assignments Exercises Examples 6. (x2 ⫺ 10x ⫺ 24) ⫼ (x ⫹ 2) x
12 7. (3a4 ⫺ 6a3 ⫺ 2a2 ⫹ a ⫺ 6) ⫼ (a ⫹ 1)
Exercises 15–50 are structured 4, 5 1 8. (z5 ⫺ 3z2 ⫺ 20) ⫼ (z ⫺ 2) 9. (x3 ⫹ y3) ⫼ (x ⫹ y) x 2
xy  y 2
6–10 2, 4
so that students practice the x3 ⫹ 13x2 ⫺ 12x ⫺ 8
11, 14 3 10. ᎏᎏᎏ 11. (b4 ⫺ 2b3 ⫹ b2 ⫺ 3b ⫹ 2)(b ⫺ 2)⫺1
x⫹2
same concepts whether they 12,13 5 9b ⫹ 9b ⫺ 10
2
12. (12y2 ⫹ 36y ⫹ 15) ⫼ (6y ⫹ 3) 13. ᎏᎏ 3b  5
are assigned odd or even 3b ⫺ 2
2y  5
problems. Standardized 14. Which expression is equal to (x2 ⫺ 4x ⫹ 6)(x ⫺ 3)⫺1? B
Test Practice 3
Assignment Guide A x⫺1 B x ⫺ 1 ⫹ ᎏᎏ
x⫺3
Basic: 15–43 odd, 49, 51, 53, C x ⫺ 1 ⫺ ᎏᎏ
3 D ⫺x ⫹ 1 ⫺ ᎏᎏ
3
x⫺3 x⫺3
54, 58–74 ★ indicates increased difficulty
Average: 15–51 odd, 53, 54, Practice and Apply
58–74
Advanced: 16–52 even, 55–68 Simplify.
9a3b2 ⫺ 18a2b3 5xy2 ⫺ 6y3 ⫹ 3x2y3 6y2
(optional: 69–74) 15. ᎏ ᎏ 3ab
6b 2 16. ᎏᎏᎏ 5y
ᎏᎏ  3xy2
3a2b xy x
All: Practice Quiz 1 (1–10) 17. 2c 2
3d  4d 2 17. (28c3d ⫺ 42cd2 ⫹ 56cd3) ⫼ (14cd) 18. (12mn3 ⫹ 9m2n2 ⫺ 15m2n) ⫼ (3mn)
18. 4n 2  3mn
5m 19. (2y3z ⫹ 4y2z2 ⫺ 8y4z5)(yz)⫺1 20. (a3b2 ⫺ a2b ⫹ 2a)(⫺ab)⫺1
2
236 Chapter 5 Polynomials 2y 2  4yz
8y 3z 4
a 2b  a
ᎏᎏ
b
FIND THE ERROR
Suggest that
students recheck their
Differentiated Instruction
calculations immediately
whenever they begin to get large Interpersonal To help discover confusions and catch careless errors,
numbers as the coefficients of have students work in pairs as they do division problems. One person
their quotient. While nothing should write the solution steps, explaining each step out loud while the
forbids large coefficients, this is other person watches, listens, and checks the work. The students should
sometimes the first indication then exchange roles and repeat the activity.
that they have made an error in
their calculations.
236 Chapter 5 Polynomials
Homework Help 21. (b3  8b2  20b)  (b  2) 22. (x2  12x  45)  (x  3)
NAME ______________________________________________ DATE ____________ PERIOD _____

Study
5-3 Guide
Study andIntervention
Guide and Intervention,
For See 23. (n3  2n2  5n  12)  (n  4) 24. (2c3  3c2  3c  4)  (c  2) p. 251 (shown)
Dividing Polynomials and p. 252
Exercises Examples
25. (x4  3x3  x2  5)  (x  2) 26. (6w5  18w2  120)  (w  2)
Use Long Division To divide a polynomial by a monomial, use the properties of powers
15–20, 51 1 from Lesson 5-1.
To divide a polynomial by a polynomial, use a long division pattern. Remember that only
21–34, 49, 2, 4
50, 52–54 27. (x3  4x2)  (x  4) 28. (x3  27)  (x  3) like terms can be added or subtracted.

Example 1 Simplify  12p3t2r  21p2qtr2  9p3tr

35–38 3, 4 y3  3y2  5y  4 m  3m  7m  21
3 2 3p2tr
.

29.  30.  m3

12p3t2r  21p2qtr2  9p3tr
 
12p3t2r 21p2qtr2
  
9p3tr
39–48 2, 3, 5 y4 2
3p tr 3p2tr 3p2tr 3p2tr
12 21 9
  p3  2t2  1r1  1   p2  2qt1  1r2  1   p3  2t1  1r1  1
3 3 3
a  5a  13a  10
4 3 2 2m  5m  10m  8
4 3
Extra Practice 31. 
a1
32. 
m3 Example 2
 4pt 7qr  3p

Use long division to find (x3  8x2  4x  9)

(x  4).
See page 837.
x5  7x3  x  1 3c5  5c4  c  5 x2  4x  12

21–48. See pp. 283A– 33.  34.  x  4

x
3
8
x
2 
4
x
9
x3 c2 ()x3  4x2
4x2  4x

283B.  8g  15)(g  3)1   2b  3)(b  1)1

()4x2  16x
35. (g2 36. (2b3 b2 12x  9
()12x  48

37. (t5  3t2  20)(t  2)1 38. (y5  32)(y  2)1 57
The quotient is x2  4x  12, and the remainder is 57.
x3  8x2  4x  9 57

Lesson 5-3
Therefore   x2  4x  12   .
39. (6t3  5t2  9)  (2t  3) 40. (2h3  5h2  22h)  (2h  3) x4 x4

Exercises
9d  5d  8
3 4x  5x  3x  1
3 2
41.  42.  Simplify.
3d  2 4x  1 1. 
18a3  30a2
3a
24mn6  40m2n3
2. 
2 3
60a2b3  48b4  84a5b2
3. 
2
4m n 12ab

   3x  6
2x4 3x3 2x2    3x  1
6x4 5x3 x2 6n 3 4b2
43.  44.  6a 2  10a   10 5ab    7a 4
2x  3 3x  1
m a
4. (2x2  5x  3)  (x  3) 5. (m2  3m  7)  (m  2)

x3  3x2  x  3 x4  x2  3x  5 2x  1 m5
3

★ 45.  ★ 46. 
m2

x2  1 x2  2 6. (p3  6)  (p  1) 7. (t3  6t2  1)  (t  2)

5 31
p2  p  1  
p1
t 2  8t  16  
t2
x3  3x2  3x  2
★ 47.  x3  4x2  5x  6
★ 48. 
  8. (x5  1)  (x  1) 9. (2x3  5x2  4x  4)  (x  2)

x2  x  1 x2  x  2 x4  x3  x2  x  1 2x 2  x  2

NAME ______________________________________________
251 DATE ____________
GlPERIOD
Al _____

49. What is x3  2x2  4x  3 divided by x  1? x 2  x  3

Gl /M G Hill b 2

Skills
5-3 Practice,
Practice (Average)
p. 253 and
Practice, p. 254 (shown)
Dividing Polynomials
50. Divide 2y3  y2  5y  2 by y  2. 2y 2  3y  1 Simplify.
15r10  5r8  40r2 8 6k2m  12k3m2  9m3 3k 9m
1.  4 3r 6  r 4  2 2. 
2   6k 2  
5r r 2km m 2k

51. BUSINESS A company estimates that it costs 0.03x2  4x  1000 dollars to 3. (30x3y  12x2y2  18x2y)  (6x2y) 4. (6w3z4  3w2z5  4w  5z)  (2w2z)
produce x units of a product. Find an expression for the average cost per unit. 5x  2y  3
3z 2
3wz 3      2
2
4

wz
5
2w
5. (4a3  8a2  a2)(4a)1 6. (28d 3k2  d 2k2  4dk2)(4dk2)1
Cost Analyst a 2  2a  
a d
7d 2    1
52. ENTERTAINMENT A magician gives these instructions to a volunteer. 4 4
Cost analysts study and f 2  7f  10
7.  f  5
f2
2x2  3x  14
8.  2x  7
x2

write reports about the • Choose a number and multiply it by 3. 9. (a3  64)  (a  4) a 2  4a  16 10. (b3  27)  (b  3) b 2  3b  9
factors involved in the
cost of production.
• Then add the sum of your number and 8 to the product you found. 72
2x3  6x  152 2x  4x  6
3
11.  2x 2  8x  38 12.  2x 2  6x  22  
x4 x3 x3
• Now divide by the sum of your number and 2.
Online Research 13. (3w3  7w2  4w  3)  (w  3) 14. (6y4  15y3  28y  6)  (y  2)
What number will the volunteer always have at the end? Explain. 3w 2  2w  2  
3
6y 3  3y 2  6y  16  
26
For information w3 y2
4; See margin for explanation. 15. (x4  3x3  11x2  3x  10)  (x  5) 16. (3m5  m  1)  (m  1)
about a career in x3  2x 2  x  2 3m4  3m 3  3m 2  3m  4  
5

cost analysis, visit: MEDICINE For Exercises 53 and 54, use the following information. 17. (x4  3x3  5x  6)(x  2)1 18. (6y2  5y  15)(2y  3)1
m1

www.algebra2.com/ The number of students at a large high school who will catch the flu during x3  5x 2
24
 10x  15  
x2
3y  7  
6
2y  3
170t2 4x2  2x  6 6x2  x  7
careers an outbreak can be estimated by n  
2 , where t is the number of
19. 
2x  3
20. 
3x  1
t 1
12 6
2x  2   2x  1  
2x  3 3x  1
weeks from the beginning of the epidemic and n is the number of ill people. 21. (2r3  5r2  2r  15)  (2r  3)
r 2  4r  5
22. (6t3  5t2  2t  1)  (3t  1)
2t 2  t  1  
2
170t2 170 3t  1
53. Perform the division indicated by  . 170    23. 
4p4  17p2  14p  3 2h4  h3  h2  h  3
24. 

51. $0.03x  4 
1000 t2  1 t2  1 2p  3
2p 3  3p 2  4p  1
2 h 1
2h 2  h  3
x 54. Use the formula to estimate how many people will become ill during the first week. 25. GEOMETRY The area of a rectangle is 2x2  11x  15 square feet. The length of the
rectangle is 2x  5 feet. What is the width of the rectangle? x  3 ft
85 people
26. GEOMETRY The area of a triangle is 15x4  3x3  4x2  x  3 square meters. The
PHYSICS For Exercises 55–57, suppose an object moves in a straight line so that length of the base of the triangle is 6x2  2 meters. What is the height of the triangle?
5x 2  x  3 m
after t seconds, it is t3  t2  6t feet from its starting point. 55. x3  x2  6x  24 ft Gl NAME
/M G ______________________________________________
Hill 254 DATE ____________
Gl PERIOD
Al _____
b 2
Reading
Readingto
to Learn
55. Find the distance the object travels between the times t  2 and t  x. 5-3 Learn Mathematics
ELL
Mathematics, p. 255
Dividing Polynomials
56. How much time elapses between t  2 and t  x? x  2 s Pre-Activity How can you use division of polynomials in manufacturing?
Read the introduction to Lesson 5-3 at the top of page 233 in your textbook.
57. Find a simplified expression for the average speed of the object between times Using the division symbol (), write the division problem that you would

t  2 and t  x. x 2  3x  12 ft/s use to answer the question asked in the introduction. (Do not actually
divide.) (32x2  x)
(8x)

58. Sample answer: Reading the Lesson

58. CRITICAL THINKING Suppose the result of dividing one polynomial by
r 3  9r 2  27r  28
1. a. Explain in words how to divide a polynomial by a monomial. Divide each term of
the polynomial by the monomial.
1
and r  3 another is r2  6r  9  . What two polynomials might have been divided?
r3
b. If you divide a trinomial by a monomial and get a polynomial, what kind of
polynomial will the quotient be? trinomial

2. Look at the following division example that uses the division algorithm for polynomials.
2x  4
www.algebra2.com/self_check_quiz Lesson 5-3 Dividing Polynomials 237 x  4
2x2  4x  7
2x2  8x
4x  7
4x  16
23
Which of the following is the correct way to write the quotient? C
NAME ______________________________________________ DATE ____________ PERIOD _____ 23 23
Answer A. 2x  4 B. x  4 C. 2x  4  
x4
D. 
x4

Enrichment, p. 256 Lesson 5-3

5-3 Enrichment
3. If you use synthetic division to divide x3  3x2  5x  8 by x  2, the division will look
like this:

52. Let x be the number. Multiplying by 3 Oblique Asymptotes 2 1 3

2
5
10
8
10

results in 3x. The sum of the number, 8, The graph of y  ax  b, where a  0, is called an oblique asymptote of y  f(x)
if the graph of f comes closer and closer to the line as x → ∞ or x → ∞. ∞ is the
1 5 5 2
Which of the following is the answer for this division problem? B
mathematical symbol for infinity, which means endless.
and the result of the multiplication is 2
For f(x)  3x  4  , y  3x  4 is an oblique asymptote because
x
A. x2  5x  5
2
B. x2  5x  5  
2
x2
C. x3  5x2  5x   D. x3  5x2  5x  2
x  8  3x or 4x  8. Dividing by the sum
2 2
f(x)  3x  4  , and  → 0 as x → ∞ or ∞. In other words, as | x | x2
x x
2
increases, the value of  gets smaller and smaller approaching 0.
4x  8
x
Helping You Remember
of the number and 2 gives  or 4. Example x2  8x  15 4. When you translate the numbers in the last row of a synthetic division into the quotient
x2
Find the oblique asymptote for f(x)  .
x2 and remainder, what is an easy way to remember which exponents to use in writing the
2 1 8 15 Use synthetic division.
terms of the quotient? Sample answer: Start with the power that is one less
than the degree of the dividend. Decrease the power by one for each
The end result is always 4. 1
2
6
12
3
term after the first. The final number will be the remainder. Drop any term
that is represented by a 0.
x2  8x  15 3
y    x  6  
x2 x2
3

Lesson 5-3 Dividing Polynomials 237

 59. WRITING IN MATH Answer the question that was posed at the beginning
4 Assess of the lesson. See pp. 283A–283B.
How can you use division of polynomials in manufacturing?
Open-Ended Assessment Include the following in your answer:
• the dimensions of the piece of metal that the machinist needs,
Writing Have students write • the formula from geometry that applies to this situation, and
their own list of tips for how to • an explanation of how to use division of polynomials to find the width s
do division problems, describing of the seam.
the techniques they use to help
avoid making errors. Standardized 60. An office employs x women and 3 men. What is the ratio of the total number of
Test Practice employees to the number of women? A
3 x 3 x
A 1   B  C  D 
x x3 x 3

Intervention 61. If a  b  c and a  b, then all of the following are true EXCEPT D
New Some students A a  c  b  c. B a  b  0.
may have C 2a  2b  2c. D c  b  2a.
trouble keeping
their concentra- Maintain Your Skills
tion throughout the sequence
Mixed Review Simplify. (Lesson 5-2) 62. x 2  4x  14 63. y 4z 4  y 3z 3  3y 2z
of steps required in long
62. (2x2  3x  5)  (3x2  x  9) 63. y2z(y2z3  yz2  3)
division. Encourage them to
compare intermediate results 64. (y  5)(y  3) y2  2y  15 65. (a  b)2 a 2  2ab  b2
with a partner, so that they 66. ASTRONOMY Earth is an average of 1.5
1011 meters from the Sun. Light
can ask questions and catch travels at 3
108 meters per second. About how long does it take sunlight to
errors before completing the reach Earth? (Lesson 5-1) 5  102 s or 8 min 20 s
entire problem. Write an equation in slope-intercept form for each graph. (Lesson 2-4)
2 4
67. y y  x  2 68. y y  x  
3 3
(2, 0)
Getting Ready for (1, 1) O x
Lesson 5-4 O x

BASIC SKILL Lesson 5-4 presents (3, 1)

(4, 4)
factoring polynomials. This re-
quires a knowledge of the greatest Getting Ready for BASIC SKILL Find the greatest common factor of each set of numbers.
common factor. Exercises 69–74 the Next Lesson 69. 18, 27 9 70. 24, 84 12 71. 16, 28 4
should be used to determine your 72. 12, 27, 48 3 73. 12, 30, 54 6 74. 15, 30, 65 5
students’ familiarity with the
greatest common factor of a set Lessons 5-1 through 5-3
of numbers.
P ractice Quiz 1
Express each number in scientific notation. (Lesson 5-1)
Assessment Options 1. 653,000,000 6.53  108 2. 0.0072 7.2  103

Practice Quiz 1 The quiz Simplify. (Lessons 5-1 and 5-2)

a6 b2 c a3 2x2
provides students with a brief 3. (3x2y)3(2x)2 108x 8y 3 4.   
a3 b2 c 4 b4c3
 xz 
x2z
5.  
4

z6
review of the concepts and skills
in Lessons 5-1 through 5-3. 6. (9x  2y)  (7x  3y) 2x  5y 7. (t  2)(3t  4) 3t 2  2t  8 8. (n  2)(n2  3n  1)
19 n3  n2  5n  2
Lesson numbers are given to the Simplify. (Lesson 5-3) 9. m 2  3  
m4 2d  d  9d  9
3 2
right of exercises or instruction 9. (m3  4m2  3m  7)  (m  4) 10.  d 2  d  3
2d  3
lines so students can review
concepts not yet mastered.
238 Chapter 5 Polynomials
Quiz (Lessons 5-1 through 5-3)
is available on p. 307 of the
Chapter 5 Resource Masters.

238 Chapter 5 Polynomials

 Factoring Polynomials Lesson
Notes

• Factor polynomials.
• Simplify polynomial quotients by factoring.
1 Focus
does factoring apply to geometry?
Suppose the expression 4x2  10x  6 represents ? units 5-Minute Check
the area of a rectangle. Factoring can be used to Transparency 5-4 Use as a
find possible dimensions of the rectangle. A  4x 2  10x  6 units2 ? units quiz or review of Lesson 5-3.

Mathematical Background notes

are available for this lesson on
FACTOR POLYNOMIALS Whole numbers are factored using prime numbers. p. 220D.
For example, 100  2  2  5  5. Many polynomials can also be factored. Their factors,
however, are other polynomials. Polynomials that cannot be factored are called
prime.
Building on Prior
Knowledge
The table below summarizes the most common factoring techniques used with
polynomials. In this lesson, students will need
to recall how to find the area of a
Factoring Techniques rectangle, and they will also
need to remember the set of
Number of Terms Factoring Technique General Case
prime numbers.
any number Greatest Common Factor (GCF) a3b2  2a2b  4ab2  ab(a2b  2a  4b)
two Difference of Two Squares a2  b2  (a  b)(a  b) does factoring apply to
Sum of Two Cubes a3  b3  (a  b)(a2  ab  b2)
Difference of Two Cubes a3  b3  (a  b)(a2  ab  b2)
geometry?
three Perfect Square Trinomials a2  2ab  b2  (a  b)2 Ask students:
a2  2ab  b2  (a  b)2
• How do Examples 1 and 2 in
General Trinomials acx2  (ad  bc)x  bd  (ax  b)(cx  d) Lesson 5-3 relate to factoring,
four or more Grouping ax  bx  ay  by  x(a  b)  y(a  b) the topic of this lesson? The
 (a  b)(x  y)
quotient and the divisor are factors
of the dividend.
Whenever you factor a polynomial, always look for a common factor first. Then
determine whether the resulting polynomial factor can be factored again using one
or more of the methods listed in the table above.
2 Teach
Example 1 GCF
Factor 6x2y2  2xy2  6x3y.
FACTOR POLYNOMIALS
6x2y2  2xy2  6x3y  (2  3  x  x  y  y)  (2  x  y  y)  (2  3  x  x  x  y) In-Class Example Power
Point®
 (2xy  3xy)  (2xy  y)  (2xy  3x2) The GCF is 2xy. The remaining
polynomial cannot be factored
 2xy(3xy  y  3x2) using the methods above. 1 Factor 10a3b2  15a2b  5ab3.
Check this result by finding the product.
5ab(2a2b  3a  b2)

A GCF is also used in grouping to factor a polynomial of four or more terms.

Lesson 5-4 Factoring Polynomials 239

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 5 Resource Masters Graphing Calculator and 5-Minute Check Transparency 5-4
• Study Guide and Intervention, pp. 257–258 Spreadsheet Masters, p. 35 Answer Key Transparencies
• Skills Practice, p. 259 Teaching Algebra With Manipulatives
• Practice, p. 260 Masters, p. 235 Technology
• Reading to Learn Mathematics, p. 261 Alge2PASS: Tutorial Plus, Lesson 8
• Enrichment, p. 262 Interactive Chalkboard

Lesson x-x Lesson Title 239

In-Class Example Power Example 2 Grouping
Point®
Factor a3  4a2  3a  12.
2 Factor x3  5x2  2x  10. a3  4a2  3a  12  (a3  4a2)  (3a  12) Group to find a GCF.
(x  5)(x 2  2)  a2(a  4)  3(a  4) Factor the GCF of each binomial.
 (a  4)(a2  3) Distributive Property
Teaching Tip Point out to stu-
dents that it is often difficult to
recognize that grouping can be You can use algebra tiles to model factoring a polynomial.
used to factor a polynomial.
Stress that this technique should
only be considered when trying
Factoring Trinomials
to factor a polynomial with four
terms. Use algebra tiles to factor 2x2  7x  3.
Study Tip Model and Analyze
Algebra Tiles • Use algebra tiles to model 2x2  7x  3.
2 2
When modeling a • To find the product that resulted in this x x x
polynomial with algebra
polynomial, arrange the tiles to form a
tiles, it is easiest to
rectangle. x x 1
arrange the x2 tiles first,
x x 1
then the x tiles and finally
the 1 tiles to form a x x 1
rectangle.
• Notice that the total area can be expressed
2 2
as the sum of the areas of two smaller x x x 2x 2  x
rectangles.
x x 1
x x 1 6x  3
x x 1

Use these expressions to rewrite the trinomial. Then factor.

2x2  7x  3  (2x2  x)  (6x  3) total area  sum of areas of smaller rectangles
 x(2x  1)  3(2x  1) Factor out each GCF.
 (2x  1)(x  3) Distributive Property

Make a Conjecture
Study the factorization of 2x2  7x  3 above.
1. What are the coefficients of the two x terms in (2x2  x)  (6x  3)? Find
their sum and their product. 1 and 6; 7; 6
2. Compare the sum you found in Exercise 1 to the coefficient of the x term
in 2x2  7x  3. They are the same.
3. 6; It is the same. 3. Find the product of the coefficient of the x2 term and the constant term
in 2x2  7x  3. How does it compare to the product in Exercise 1?
4. Find two numbers 4. Make a conjecture about how to factor 3x2  7x  2.
with a product of
3  2 or 6 and a sum
of 7. Use those The FOIL method can help you factor a polynomial into the product of two
numbers to rewrite binomials. Study the following example.
the trinomial. Then
factor. F O I L




(ax  b)(cx  d)  ax  cx  ax  d  b  cx  b  d
 acx2  (ad  bc)x  bd

Notice that the product of the coefficient of x2 and the constant term is abcd. The
product of the two terms in the coefficient of x is also abcd.
240 Chapter 5 Polynomials

Algebra Activity
Materials: algebra tiles
• Inform students that the two factors of the trinomial can be read directly from
the completed array of tiles in another way. Point out that the length of the
array is 2x  1, and the height is x  3. The product of the length and width
gives the area, 2x2  7x  3, of the array.
• You might wish to have students experiment to see if there is another way to
form a rectangle with the tiles.

240 Chapter 5 Polynomials

 Example 3 Two or Three Terms In-Class Example Power
Point®
Factor each polynomial.
a. 5x2  13x  6 3 Factor each polynomial.
To find the coefficients of the x-terms, you must find two numbers whose a. 3y2  2y  5 (3y  5)(y  1)
product is 5  6 or 30, and whose sum is 13. The two coefficients must be
10 and 3 since (10)(3)  30 and 10  (3)  13. b. 5mp2  45m 5m(p  3)(p  3)
Rewrite the expression using 10x and 3x in place of 13x and factor c. x3y3  8
by grouping.
(xy  2)(x2y2  2xy  4)
5x2  13x  6  5x2  10x  3x  6 Substitute 10x  3x for 13x.
d. 64x6  y6 (2x  y)(4x2  2xy 
 (5x2  10x)  (3x  6) Associative Property
y 2)(2x  y)(4x2  2xy  y 2)
 5x(x  2)  3(x  2) Factor out the GCF of each group.
 (5x  3)(x  2) Distributive Property Teaching Tip Emphasize the
importance of checking each
b. 3xy2  48x
factor to make sure it is prime
3xy2  48x  3x(y2  16) Factor out the GCF. before deciding that the final
 3x(y  4)(y  4) y2  16 is the difference of two squares. group of factors has been found.
c. c3d3  27
c3d3  (cd)3 and 27  33. Thus, this is the sum of two cubes.
c3d3  27  (cd  3)[(cd)2  3(cd)  32] Sum of two cubes formula with a  cd and b  3
Concept Check
 (cd  3)(c2d2  3cd  9) Simplify.
Ask students to write an ordered
d. m6  n6 list describing what they will
This polynomial could be considered the difference of two squares or the check for as they factor a poly-
difference of two cubes. The difference of two squares should always be done nomial. Sample answer: First look
before the difference of two cubes. This will make the next step of the
factorization easier. for any common factors of the terms;
m6  n6  (m3  n3)(m3  n3) Difference of two squares
if there is a common factor, find the
GCF. After factoring out the GCF, look
 (m  n)(m2  mn  n2)(m  n)(m2  mn  n2) Sum and difference
of two cubes for the difference of two squares, a
perfect square trinomial, and so on.
You can use a graphing calculator to check that the factored form of a polynomial Use the factoring techniques listed on
is correct. p. 239 to factor the expression further.
Examine the resulting factored form to
see if there are any factors that are not
Factoring Polynomials prime. If so, continue the process. If
Is the factored form of 2x2  11x  21 not, the factoring is complete.
equal to (2x  7)(x  3)? You can find
out by graphing y  2x2  11x  21 and
y  (2x  7)(x  3). If the two graphs coincide,
the factored form is probably correct.
• Enter y  2x2  11x  21 and
y  (2x  7)(x  3) on the Y= screen.
• Graph the functions. Since two different graphs [10, 10] scl: 1 by [40, 10] scl: 5
appear, 2x2  11x  21  (2x  7)(x  3).
Think and Discuss
2. No; in some cases, 1. Determine if x2  5x  6  (x  3)(x  2) is a true statement. If not, write
the graphs might be the correct factorization. no; (x  6)(x  1)
so close in shape that 2. Does this method guarantee a way to check the factored form of a
they seem to coincide polynomial? Why or why not?
but do not.

www.algebra2.com/extra_examples Lesson 5-4 Factoring Polynomials 241

Factoring Polynomials So that students see what happens when a

polynomial and a correct factorization are graphed, have students graph the
functions y  x2  81 and y  (x  9)(x  9) in the same screen. It looks
like only one graph appears on the screen because both graphs are the same.

Lesson 5-4 Factoring Polynomials 241

SIMPLIFY QUOTIENTS SIMPLIFY QUOTIENTS In Lesson 5-3, you learned to simplify the quotient of
two polynomials by using long division or synthetic division. Some quotients can be
In-Class Example Power simplified using factoring.
Point®

a a6 2 Example 4 Quotient of Two Trinomials

4 Simplify  .
a2  7a  10 x2  2x  3
Simplify  .
a3 2 x  7x  12
 , if a  5, 2 1
a5 x2  2x  3 (x  3)(x  1)
    Factor the numerator and denominator.
x2  7x  12 (x  4)(x  3)
1
x1
  Divide. Assume x  3, 4.
x4
3 Practice/Apply Therefore, 
2
x2  2x  3
x  7x  1 2
x1
  , if x  3, 4.
x4

Study Notebook
Have students—
• add the definitions/examples of Concept Check 1. OPEN ENDED Write an example of a perfect square trinomial.
the vocabulary terms to their 1. Sample answer: 2. Find a counterexample to the statement a2  b2  (a  b)2.
Vocabulary Builder worksheets for x2  2x  1 x2 1
3. Decide whether the statement     is sometimes, always, or
x2  x  6 x3
Chapter 5. never true. sometimes
• add a list of factoring techniques 2. Sample answer: If a  1 and b  1, then a2  b2  2 but (a  b)2  4.
Guided Practice Factor completely. If the polynomial is not factorable, write prime.
to their notebook, including the 4. 12x2  6x 6x(2x  1) 5. a2  5a  ab a(a  5  b)
GUIDED PRACTICE KEY
factoring of the special cases Exercises Examples 6. 21  7y  3x  xy (x  7)(3  y) 7. y2  6y  8 (y  2)(y  4)
listed in the Concept Summary on 4–5 1 8. z2  4z  12 (z  6)(z  2) 9. 3b2  48 3(b  4)(b  4)
p. 239 and the FOIL method 6 2 10. 16w2  169 (4w  13)(4w  13) 11. h3  8000 (h  20)(h2  20h  400)
7–11, 14 3
described on p. 240. 12–13 4 Simplify. Assume that no denominator is equal to 0.
• include any other item(s) that they x2  2x  8 x  4 2y2  8y 2y
12.    13.   
find helpful in mastering the skills x2  5x  14 x  7 y2  16 y  4

in this lesson. Application 14. GEOMETRY Find the width of rectangle ABCD 3x  6y cm
A D
if its area is 3x2  9xy  6y2 square centimeters.
x  y cm

B C

★ indicates increased difficulty

About the Exercises…
Organization by Objective
Practice and Apply
• Factor Polynomials: 15–45 Factor completely. If the polynomial is not factorable, write prime.
• Simplify Quotients: 46–51 15. 2xy3  10x 2x(y3  5) 16. 6a2b2  18ab3 6ab2(a  3b)
Odd/Even Assignments 17. 2cd 2(6d  4c  17. 12cd3  8c2d2  10c5d3 18. 3a2bx  15cx2y  25ad3y prime
5c 4d) 19. 8yz  6z  12y  9 (2z  3)(4y  3) 20. 3ax  15a  x  5 (3a  1)(x  5)
Exercises 15–44 and 46–51 are
structured so that students 21. x2  7x  6 (x  1)(x  6) 22. y2  5y  4 (y  1)(y  4)
practice the same concepts 23. 2a2  3a  1 (2a  1)(a  1) 24. 2b2  13b  7 (2b  1)(b  7)
whether they are assigned 25. 6c2  13c  6 (2c  3)(3c  2) 26. 12m2  m  6 (3m  2)(4m  3)
odd or even problems. 27. 3n2  21n  24 3(n  8)(n  1) 28. 3z2  24z  45 3(z  3)(z  5)
242 Chapter 5 Polynomials
Assignment Guide
Basic: 15–37 odd, 43–49 odd,
55–58, 63–81
Differentiated Instruction
Average: 15–51 odd, 55–58,
63–81 (optional: 59–62) Auditory/Musical Ask students to create songs or raps to help them
remember the factoring techniques for the difference of two squares, for
Advanced: 16–50 even, 52–75 the sum or difference of two cubes, or for one of the two perfect square
(optional: 76–81) trinomial types.

242 Chapter 5 Polynomials

 29. x2  12x  36 (x  6)2 30. x2  6x  9 (x  3)2
NAME ______________________________________________ DATE ____________ PERIOD _____
Homework Help Study
5-4 Guide
Study andIntervention
Guide and Intervention,
For See 31. 16a2  25b2 prime 32. 3m2  3n2 3(m  n)(m  n) p. 257 (shown)
Factoring Polynomialsand p. 258
Exercises Examples
Factor Polynomials
15–18 1 33. y4  z2 (y2  z)(y2  z) 34. 3x2  27y2 3(x  3y)(x  3y) For any number of terms, check for:

19, 20 2
35. z3  125 (z  5)(z2  5z  25) 36. t3  8 (t  2)(t 2  2t  4)
greatest common factor
For two terms, check for:
21–38, 3 Difference of two squares

37. p4  1 (p2  1)(p  1)(p  1) 38. x4  81 (x2  9)(x  3)(x  3)

a 2  b 2  (a  b)(a  b)
43–45, 55 Sum of two cubes
a 3  b 3  (a  b)(a 2  ab  b 2)
39–42 2, 3
★ 39. 7ac2  2bc2  7ad2  2bd2 (7a  2b)(c  d)(c  d)
Difference of two cubes
a 3  b 3  (a  b)(a 2  ab  b 2)
46–54 4 Techniques for Factoring Polynomials For three terms, check for:

★ 40. 8x2  8xy  8xz  3x  3y  3z (8x  3)(x  y  z)

Perfect square trinomials
a 2  2ab  b 2  (a  b)2
a 2  2ab  b 2  (a  b)2
Extra Practice ★ 41. 5a2x  4aby  3acz  5abx  4b2y  3bcz (a  b)(5ax  4by  3cz)
General trinomials
acx 2  (ad  bc)x  bd  (ax  b)(cx  d)
See page 837. For four terms, check for:

★ 42. 3a3  2a2  5a  9a2b  6ab  15b (a  3b)(3a  5)(a  1)

Grouping
ax  bx  ay  by  x(a  b)  y(a  b)
 (a  b)(x  y)

43. Find the factorization of 3x2  x  2. (3x  2)(x  1) Example Factor 24x2  42x  45.
First factor out the GCF to get 24x2  42x  45  3(8x2  14x  15). To find the coefficients
of the x terms, you must find two numbers whose product is 8  (15)  120 and whose

44. What are the factors of 2y2  9y  4? (2y  1)(y  4) sum is 14. The two coefficients must be 20 and 6. Rewrite the expression using 20x and
6x and factor by grouping.
8x2  14x  15  8x2  20x  6x  15 Group to find a GCF.
 4x(2x  5)  3(2x  5) Factor the GCF of each binomial.
 (4x  3)(2x  5) Distributive Property

Thus, 24x2  42x  45  3(4x  3)(2x  5).

45. LANDSCAPING A boardwalk that is x feet x

Lesson 5-4
Exercises
wide is built around a rectangular pond. The
Factor completely. If the polynomial is not factorable, write prime.
combined area of the pond and the boardwalk a
1. 14x2y2  42xy3 2. 6mn  18m  n  3 3. 2x2  18x  16
is 4x2  140x  1200 square feet. What are the 14xy 2(x  3y) (6m  1)(n  3) 2(x  8)(x  1)

dimensions of the pond? 30 ft by 40 ft 4. x4 1 5. 35x3y4  60x4y 6. 2r3  250

(x 2  1)(x  1)(x  1) 5x 3y(7y 3  12x) 2(r  5)(r 2  5r  25)

7. 100m8  9 8. x2  x  1 9. c4  c3  c2  c
(10m 4  3)(10m 4  3) prime c(c  1)2 (c  1)

Gl NAME
/M G ______________________________________________
Hill 257 DATE ____________
GlPERIOD
Al _____
b 2

Simplify. Assume that no denominator is equal to 0. Skills

5-4 Practice,
Practice (Average)
p. 259 and
Practice, p. 260 (shown)
Factoring Polynomials
x2  4x  3 x  1 x2  4x  5 x  5
46.    47.    Factor completely. If the polynomial is not factorable, write prime.

x2  x  12 x  4 x2  7x  6 x  6 1. 15a2b  10ab2 2. 3st2  9s3t  6s2t2 3. 3x3y2  2x2y  5xy

5ab(3a  2b) 3st(t  3s 2  2st) xy(3x 2y  2x  5)
x  25
2 x5 x  6x  8
2 x4
48.    49.     4. 2x3y  x2y  5xy2  xy3 5. 21  7t  3r  rt 6. x2  xy  2x  2y
x2  3x  10 x  2 x3  8 x2  2x  4 xy(2x 2  x  5y  y 2) (7  r)(3  t) (x  2)(x  y)
7. y2  20y  96 8. 4ab  2a  6b  3 9. 6n2  11n  2
x2 x1
★ 50.  
(x2  x)(x  1)1
x ★ 51. 
(x2  3x  2)(x  2)2
x2
(y  8)(y  12)
10. 6x2  7x  3
(2a  3)(2b  1)
11. x2  8x  8
(6n  1)(n  2)
12. 6p2  17p  45
(3x  1)(2x  3) prime (2p  9)(3p  5)
13. r3  3r2  54r 14. 8a2  2a  6 15. c2  49
r(r  9)(r  6) 2(4a  3)(a  1) (c  7)(c  7)
BUILDINGS For Exercises 52 and 53, use the following information. 16. x3  8 17. 16r2  169 18. b4  81
When an object is dropped from a tall building, the distance it falls between (x  2)(x 2  2x  4) (4r  13)(4r  13) (b 2  9)(b  3)(b  3)

1 second after it is dropped and x seconds after it is dropped is 16x2  16 feet. 19. 8m3  25 prime 20. 2t3  32t2  128t 2t(t  8)2

21. 5y5  135y2 5y 2(y  3)(y 2  3y  9) 22. 81x4  16 (9x 2  4)(3x  2)(3x  2)
52. How much time elapses between 1 second after it is dropped and x seconds
after it is dropped? x  1 s Simplify. Assume that no denominator is equal to 0.
x  16 x4 x  16x  64  x8 3x  27  3(x  3)
53. What is the average speed of the object during that time period? 16x  16 ft/s
2 2 2
23.   24.  25. 
2 x5
x  x  20 2x  x  72x9 3x  272 x  3x  9
26. DESIGN Bobbi Jo is using a software package to create a x cm
drawing of a cross section of a brace as shown at the right.
Write a simplified, factored expression that represents the
area of the cross section of the brace. x(20.2  x) cm2
54. GEOMETRY The length of one leg of a right triangle is x  6 centimeters,

12 cm
1
and the area is x2  7x  24 square centimeters. What is the length of the x cm
2
other leg? x  8 cm
8.2 cm

27. COMBUSTION ENGINES In an internal combustion engine, the up

and down motion of the pistons is converted into the rotary motion of r1
the crankshaft, which drives the flywheel. Let r1 represent the radius r2

55. CRITICAL THINKING Factor 64p2n  16pn  1. (8pn  1)2

of the flywheel at the right and let r2 represent the radius of the
crankshaft passing through it. If the formula for the area of a circle
is A  r2, write a simplified, factored expression for the area of the
cross section of the flywheel outside the crankshaft.  (r1  r2)(r1  r2)

Buildings Gl
Reading
NAME
/M G ______________________________________________
Hill 260 DATE ____________
Gl PERIOD
Al _____
b 2

56. WRITING IN MATH Answer the question that was posed at the beginning of 5-4 Readingto
to Learn
Learn Mathematics
The tallest buildings in the lesson. See pp. 283A–283B. Mathematics, p. 261 ELL
Factoring Polynomials
the world are the
Pre-Activity How does factoring apply to geometry?
Petronas Towers in Kuala How does factoring apply to geometry? Read the introduction to Lesson 5-4 at the top of page 239 in your textbook.
Lumpur, Malaysia. Each Include the following in your answer: If a trinomial that represents the area of a rectangle is factored into two
binomials, what might the two binomials represent? the length and
is 1483 feet tall. width of the rectangle

Source: www.worldstallest.com
• an explanation of how to use factoring to find possible dimensions for the
rectangle described at the beginning of the lesson, and Reading the Lesson
1. Name three types of binomials that it is always possible to factor. difference of two
• why your dimensions are not the only ones possible, even if you assume that squares, sum of two cubes, difference of two cubes

the dimensions are binomials with integer coefficients. 2. Name a type of trinomial that it is always possible to factor. perfect square
trinomial
www.algebra2.com/self_check_quiz Lesson 5-4 Factoring Polynomials 243
3. Complete: Since x2  y2 cannot be factored, it is an example of a prime
polynomial.

4. On an algebra quiz, Marlene needed to factor 2x2  4x  70. She wrote the following
answer: (x  5)(2x  14). When she got her quiz back, Marlene found that she did not
NAME ______________________________________________ DATE ____________ PERIOD _____ get full credit for her answer. She thought she should have gotten full credit because she
checked her work by multiplication and showed that (x  5)(2x  14)  2x2  4x  70.

Enrichment,
5-4 Enrichment p. 262 a. If you were Marlene’s teacher, how would you explain to her that her answer was not
entirely correct? Sample answer: When you are asked to factor a
polynomial, you must factor it completely. The factorization was not
complete, because 2x  14 can be factored further as 2(x  7).
Using Patterns to Factor
Study the patterns below for factoring the sum and the difference of cubes. b. What advice could Marlene’s teacher give her to avoid making the same kind of error
in factoring in the future? Sample answer: Always look for a common
a3  b3  (a  b)(a2  ab  b2)
factor first. If there is a common factor, factor it out first, and then see
Lesson 5-4

a3  b3  (a  b)(a2  ab  b2) if you can factor further.

This pattern can be extended to other odd powers. Study these examples.

Example 1 Factor a5  b5. Helping You Remember

Extend the first pattern to obtain a5  b5  (a  b)(a4  a3b  a2b2  ab3  b4). 5. Some students have trouble remembering the correct signs in the formulas for the sum
Check: (a  b)(a4  a3b  a2b2  ab3  b4)  a5  a4b  a3b2  a2b3  ab4 and difference of two cubes. What is an easy way to remember the correct signs?
 a4b  a3b2  a2b3  ab4  b5 Sample answer: In the binomial factor, the operation sign is the same as
 a5  b5 in the expression that is being factored. In the trinomial factor, the
operation sign before the middle term is the opposite of the sign in the
expression that is being factored. The sign before the last term is always
Example 2 Factor a5  b5. a plus.
Extend the second pattern to obtain a5  b5  (a  b)(a4  a3b  a2b2  ab3  b4).
Check: (a  b)(a4  a3b  a2b2  ab3  b4)  a5  a4b  a3b2  a2b3  ab4
 a4b  a3b2  a2b3  ab4  b5

Lesson 5-4 Factoring Polynomials 243

 Standardized 57. Which of the following is the factorization of 2x  15  x2? B
4 Assess Test Practice A
C
(x  3)(x  5)
(x  3)(x  5)
B
D
(x  3)(x  5)
(x  3)(x  5)
Open-Ended Assessment 58. Which is not a factor of x3  x2  2x? C
Modeling Have students use A x B x1 C x1 D x2
algebra tiles to model and
determine the factored form of Graphing CHECK FACTORING Use a graphing calculator to determine if each polynomial is
the polynomial 3x2  11x  6. Calculator factored correctly. Write yes or no. If the polynomial is not factored correctly, find
(3x  2)(x  3) the correct factorization. 60. no; (x  2)(x2  2x  4)
59. 3x2  5x  2  (3x  2)(x  1) yes 60. x3  8  (x  2)(x2  x  4)
61. 2x2  5x  3  (x  1)(2x  3) 62. 3x2  48  3(x  4)(x  4) yes
no; (2x  1)(x  3)
Intervention
New Point out that
factoring in-
volving a2  b2, Maintain Your Skills
a3  b3, a3  b3, Mixed Review Simplify. (Lesson 5-3)
(a  b) , and (a  b)2 occurs so
2
63. (t3  3t  2)  (t  2) t 2  2t  1 64. (y2  4y  3)(y  1)1 y  3
frequently in algebra that stu-
x  3x  2x  6
3 2 3x  x  8x  10x  3
4 3 2
dents should memorize these 65.  x2  2 66. 
x3 3x  2 1
forms. Then students can x3  x2  2x  2  
3x  2
easily recognize and use them Simplify. (Lesson 5-2)
whenever the need arises. 67. (3x2  2xy  y2)  (x2  5xy  4y2) 68. (2x  4)(7x  1) 14x2  26x  4
4x2  3xy  3y2
Perform the indicated operations, if possible. (Lesson 4-5)
Getting Ready for 69. [3 1] 
0
2 
[2] 70.
1 4
2 2

0 3
9 1   36
18 4
7
Lesson 5-5
71. PHOTOGRAPHY The perimeter of a rectangular picture is 86 inches. Twice the
PREREQUISITE SKILL Lesson 5-5
width exceeds the length by 2 inches. What are the dimensions of the picture?
discusses radicals. Many radicals (Lesson 3-2) 15 in. by 28 in.
are irrational numbers whose
approximate value can be found Determine whether each relation is a function. Write yes or no. (Lesson 2-1)
using a calculator. Exercises 76–81 72. y yes 73. y no
should be used to determine
your students’ familiarity with
rational and irrational numbers.
O x O x

State the property illustrated by each equation. (Lesson 1-2)

74. (3  8)5  3(5)  8(5) Distributive 75. 1  (7  4)  (1  7)  4
Associative ()
Getting Ready for PREREQUISITE SKILL Determine whether each number is rational or irrational.
the Next Lesson (To review rational and irrational numbers, see Lesson 1-2.) 80. irrational
16
76. 4.63 rational 77.  irrational 78.  rational
3
79. 8.333… rational 80. 7.323223222… 81. 9.71 rational
244 Chapter 5 Polynomials

Unlocking Misconceptions
• Difference of Two Squares Many people think that the expressions
a2  b2 and (a  b)2 are the same. Have students choose values for
a and b, such as a  5 and b  3, to see that this is not true.
• Sum of Two Squares Students may need to be convinced that
a2  b2 cannot be factored after seeing that a3  b3 can be factored.
Have them substitute values for a and b to test possible factored
forms, such as (a  b)(a  b), to verify they do not equal a2  b2.

244 Chapter 5 Polynomials

 Roots of Real Numbers Lesson
Notes

• Simplify radicals.
• Use a calculator to approximate radicals.
1 Focus
Vocabulary do square roots apply to oceanography?
• square root The speed s in knots of a wave can be 5-Minute Check

• nth root
estimated using the formula s  1.34, Transparency 5-5 Use as a
• principal root where  is the length of the wave in feet. quiz or review of Lesson 5-4.
This is an example of an equation that
contains a square root. Mathematical Background notes
are available for this lesson on
p. 220D.

do square roots apply

to oceanography?
SIMPLIFY RADICALS Finding the square root of a number and squaring
a number are inverse operations. To find the square root of a number n, you must Ask students:
find a number whose square is n. For example, 7 is a square root of 49 since 72  49. • One knot means one nautical
Since (7)2  49, 7 is also a square root of 49.
mile per hour and one nautical
mile is about 6076 feet. One mile
Definition of Square Root on land (called a statute mile) is
• Words For any real numbers a and b, if a2  b, then a is a square root of b. 5280 feet. Which is faster, 1 knot
• Example Since 52  25, 5 is a square root of 25. or 1 statute mile per hour?
1 knot
• As the length of a wave (repre-
Since finding the square root of a number and squaring a number are inverse sented by  in the diagram)
operations, it makes sense that the inverse of raising a number to the nth power is increases, does the speed of the
finding the nth root of a number. The table below shows the relationship between wave increase or decrease?
raising a number to a power and taking that root of a number. increases
Powers Factors Roots
a 3  125 5  5  5  125 5 is a cube root of 125.
a 4  81 3  3  3  3  81 3 is a fourth root of 81.
a 5  32 2  2  2  2  2  32 2 is a fifth root of 32.
an  b aaaa…ab a is an n th root of b.


n factors of a

This pattern suggests the following formal definition of an nth root.

Definition of nth Root

• Words For any real numbers a and b, and any positive integer n, if an  b,
then a is an nth root of b.
• Example Since 25  32, 2 is a fifth root of 32.

Lesson 5-5 Roots of Real Numbers 245

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 5 Resource Masters Teaching Algebra With Manipulatives 5-Minute Check Transparency 5-5
• Study Guide and Intervention, pp. 263–264 Masters, pp. 236–237 Answer Key Transparencies
• Skills Practice, p. 265
• Practice, p. 266 Technology
• Reading to Learn Mathematics, p. 267 Interactive Chalkboard
• Enrichment, p. 268
• Assessment, pp. 307, 309

Lesson x-x Lesson Title 245

 The symbol  indicates an nth root.
n
Study Tip
2 Teach Reading Math
 is read the nth
50
n
index
radical sign


50
n
root of 50. radicand
SIMPLIFY RADICALS
Some numbers have more than one real nth root. For example, 36 has two square
In-Class Example Power
Point® roots, 6 and 6. When there is more than one real root, the nonnegative root is
called the principal root . When no index is given, as in 36, the radical sign
1 Simplify. indicates the principal square root. The symbol b stands for the principal nth root
n

of b. If n is odd and b is negative, there will be no nonnegative root. In this case, the
a. 
16x6 4x 3 principal root is negative.
b. 
(q3  
5)4 (q 3  5)2 4
16 
16 indicates the principal square root of 16.
5
c. 
243a10
b15 3a2b 3 
16  4  indicates the opposite of the principal square root of 16.
16

d. 4
 not a real number 
16  4  indicates both square roots of 16.  means positive or negative.
16
  5
125  indicates the principal cube root of 125.
125
3 3

Teaching Tip Be sure students

  3
81  indicates the opposite of the principal fourth root of 81.
81
4 4
understand that since 32  9 and
(3)2  9, then the equation
x2  9 has two roots, 3 and 3. The chart below gives a summary of the real nth roots of a number b.
However, the value of the ex-
pression  9 is 3 only. To indi- Real nth roots of b, 
n
b, or 
n
b
n n
cate both square roots and not n b if b 0 b if b 0 b0
just the principal root, the ex- one positive root, one negative root no real roots
pression must be given as  9. even
  5
625  is not a real number.
4
4
one real root, 0
0  0
n
one positive root, no negative roots no positive roots, one negative root
odd
8  2 
3 5
Teaching Tip When discussing the 32  2

information following Example 1,

offer this alternative. Another way to
simplify radicals that involve only Example 1 Find Roots
numbers and no variables, is to Simplify.
simplifying the expression under the
radical sign first. For example, a. 
25x4 b. 
(y2  
2)8

 (5)2 could be rewritten by first 

25x4  
(5x2)2 
(y2  
2)8  
[(y2 
2)4]2
simplifying under the radical sign to  5x2  (y2  2)4
get  25, and then taking the The square roots of 25x4 The opposite of the principal square
principal root to get 5. Similarly, are 5x2. root of (y2  2)8 is (y2  2)4.
 (2)6 simplifies to  64, whose 5

principal square root is 8. c.   

32x15y20 
d. 9
n is even.
  
5 5
32x15y20 (2x3y4)5
 
9  9
2
 2x3y4 b is negative.
The principal fifth root of
Reading Tip
New Make sure that 32x15y20 is 2x3y4.  is not a real number.
Thus, 9
students under-
stand what it When you find the nth root of an even power and the result is an odd power, you
must take the absolute value of the result to ensure that the answer is nonnegative.
means to say
that the radical sign  
(5)2  5 or 5 
(2)6  (2)3 or 8
designates the principal root. If the result is an even power or you find the nth root of an odd power, there is no
need to take the absolute value. Why?
246 Chapter 5 Polynomials

Unlocking Misconceptions
• Variables Some students tend to think that x must represent a posi-
tive number and x must represent a negative number. Reading x as
“the opposite of x” should help them understand that x is 5 if x  5.
• Square Roots of Negative Numbers Explain that 9 has no square
root that is a real number. That is, no real number can be squared to
give 9. However, inform students that  9 does represent a number,
called an imaginary number. Lesson 5-9 discusses such numbers.

246 Chapter 5 Polynomials

 Example 2 Simplify Using Absolute Value In-Class Example Power
Point®
Simplify.
8
a. x8
4

b.  81(a 
1)12 2 Simplify.
6
Note that x is an eighth root of x8. 
81(a 
1)12  
[3(a  a. 
t6 | t |
4 4
1)3]4
The index is even, so the principal
Since the index 4 is even and the 5
root is nonnegative. Since x could
exponent 3 is odd, you must use b. 243(x
 
2)15 3(x  2)3
be negative, you must take the
absolute value of x to identify the the absolute value of (a  1)3.

81(a 
4
principal root. 1)12  3(a  1)3

8
x8  x
APPROXIMATE RADICALS
WITH A CALCULATOR
APPROXIMATE RADICALS WITH A CALCULATOR Recall that real
numbers that cannot be expressed as terminating or repeating decimals are irrational In-Class Example Power
Point®
numbers. 2 and 3 are examples of irrational numbers. Decimal approximations
for irrational numbers are often used in applications. 3 PHYSICS Use the formula
Example 3 Approximate a Square Root given in Example 3. Find the
value of T for a 1.5-foot-long
PHYSICS The time T in seconds that it takes a pendulum to make a complete pendulum. about 1.36 s
L
 g
swing back and forth is given by the formula T  2 , where L is the length
Teaching Tip Help students see
of the pendulum in feet and g is the acceleration due to gravity, 32 feet how time and the length of a
per second squared. Find the value of T for a 3-foot-long pendulum in a pendulum are related by having
grandfather clock. them experiment using weights
Study Tip
with different lengths of string.
Graphing Explore You are given the values of L and g and must find the value of T.
Calculators Since the units on g are feet per second squared, the units on the
To find a root of index time T should be seconds.
greater than 2, first type
the index. Then select
0 from the
x

MATH
Plan Substitute the values for L and g into the formula. Use a calculator
to evaluate. 3 Practice/Apply
menu. Finally, enter
the radicand.
Solve g
L
T  2  Original formula

Study Notebook

3
 2  L  3, g  32
32
 1.92 Use a calculator.
Have students—
• add the definitions/examples of
It takes the pendulum about 1.92 seconds to make a complete swing.
the vocabulary terms to their
Examine The closest square to 3 is 1, and  is approximately 3, so the answer Vocabulary Builder worksheets for
32 9
Chapter 5.
1
 1
 
should be close to 2(3)   2(3)  or 2. The answer is reasonable.
9 3 • keep a list of study tips for the
graphing calculator, including the
one in this lesson.
• include any other item(s) that they
Concept Check 1. OPEN ENDED Write a number whose principal square root and cube root are find helpful in mastering the skills
both integers. Sample answer: 64 in this lesson.
2. Explain why it is not always necessary to take the absolute value of a result to
indicate the principal root. See margin.
3. Determine whether the statement 
4
(x)4  x is sometimes, always, or never true.
Explain your reasoning. Sometimes; it is true when x 0.
www.algebra2.com/extra_examples Lesson 5-5 Roots of Real Numbers 247 Answer
2. If all of the powers in the result of
an even root have even exponents,
Differentiated Instruction the result is nonnegative without
taking absolute value.
Visual/Spatial Have students create various rectangles using index cards
or cardboard, or using masking tape on the classroom floor. Have them
use the formula d   2  w2 to find the length of the diagonal of each
rectangle. After creating several rectangles, have students experiment with
using the diagonal measures of two rectangles to create another rectangle
whose length and width are irrational numbers. Students should then
find the length of the diagonal of this new rectangle.

Lesson 5-5 Roots of Real Numbers 247

 Study
5-5 Guide
NAME ______________________________________________ DATE

andIntervention
Intervention,
____________ PERIOD _____
Guided Practice Use a calculator to approximate each value to three decimal places.
Study Guide and
 8.775
4. 77  2.668
5. 19  2.632
6. 48
3 4
p. 263 (shown)
Roots of and p. 264
Real Numbers GUIDED PRACTICE KEY
Simplify Radicals
Exercises Examples
Square Root For any real numbers a and b, if a 2  b, then a is a square root of b.
For any real numbers a and b, and any positive integer n, if a n  b, then a is an nth
Simplify. 10. not a real number 13. 6ab2 14. 4x  3y
nth Root 4–6 3
 4 8.   3 
root of b.

7. 64 9. 243 10. 4096

3 5 4
Real nth Roots of b,
1.
2.
If
If
n
n
is
is
even and b 0, then b has one positive root and one negative root.
odd and b 0, then b has one positive root. 7–14 1, 2 (2)2 2
n n
b
, b
 3. If n is even and b  0, then b has no real roots.
15 3
11.  12.  13.  14. 
(4x  
3 4
x3 x y4 y 36a2b4 3y)2
4. If n is odd and b  0, then b has one negative root.

Example 1 Simplify 
49z8. Example 2 Simplify  
(2a 
3
1)6

49z8  
(7z4)2  7z4 
3
1)6  
(2a  [(2a 
3
1)2]3  (2a  1)2
z4 must be positive, so there is no need to
take the absolute value.
Application 15. OPTICS The distance D in miles from an observer to the horizon over flat land
Exercises
or water can be estimated using the formula D  1.23h , where h is the height
Simplify. in feet of the point of observation. How far is the horizon for a person whose
1. 81

3
2. 343
 3. 
144p6 eyes are 6 feet above the ground? about 3.01 mi
9 7 12| p 3 |

4.  5. 
243p10 6. 
5 3
4a10 m6n9
2a 5 3p 2 m 2n 3

7. 
3
b12 8. 
16a10
b8 9. 
121x6 ★ indicates increased difficulty
b4 4| a 5| b4 11| x 3 |
10. 
(4k)4
16k 2
11. 
169r4
13r 2
12. 27p
6
3p 2
3
Practice and Apply
13. 
625y2
z4 14. 
36q34 15. 
100x2
y4z6
25| y | z 2 6 | q17| 10| x | y 2 | z 3| Homework Help Use a calculator to approximate each value to three decimal places.
3
18. 0.64p
10  For See
 11.358  12.124  0.933
16. 0.02
7  17. 0.36

0.3 not a real number 0.8 | p 5| Exercises Examples 16. 129 17. 147 18. 0.87
19.  20. 
(11y2)4 21. 
(5a2b)6
4 3
Lesson 5-5

 2.066
19. 4.27  3.893
20. 59  7.830
21. 480
(2x)8 16–27, 3 3 3
4x 2 121y 4 25a 4b 2 60–62
22. 
(3x  23. 
(m   24. 
36x2 
12x 
3

 4.953
22. 602  3.890
23. 891  4.004
24. 4123
1)2 5)6 1 28–59 1, 2 4 5 6
| 3x  1| (m  5)2 | 6x  1|

 4.647
25. 46,815 26.  2 59.161
27. (3500)
7 6 4
Gl

Skills
5-5 Practice,
Practice
NAME
/M G

(Average)
______________________________________________
Hill

p. 265 and
263 DATE ____________
GlPERIOD
Al _____
b 2
Extra Practice (723)3 26.889
Practice, p. Numbers
266 (shown) See page 838.
Roots of Real
Use a calculator to approximate each value to three decimal places. Simplify.
 15
28. 225  13
29. 169 2
30. (7)
3 3
1. 7.8
 2. 89
 3. 25
 4. 4

2.793 9.434 2.924 1.587 30. not a real number
8. (0.94)
2
4 5 6 4
5. 1.1
 6. 0.1
 7. 5555

31.   3
32. 27  2
33. 128
3 7
1.024 0.631 4.208 0.970 (18)2 18

 
Simplify.
1 1 3 1 1
9. 0.81
 10. 324

4
11. 256

6
12. 64
 34.  
16 4
35.  
125 5
 0.5
36. 0.25
0.9 18 4 2

4 0.4
37. 0.06 38.  39. 
3 4 6
z8 z 2 x6 x
3 3 5 4
13. 64
 14. 0.512
 15. 243
 16. 1296

4 0.8 3 6
17. 
5 1024

243
18. 
243x10
5
19. 
(14a)2 20. (14a
 )2 not a
real number 40. 
49m6 7m3 41. 
64a8 8a4 42. 
3
27r3 3r
14| a|
4
 3x 2


3

43.  44.  45. 

3 3
c6 c2 (5g)4 25g2 (2z)6 4z2
16m2
21. 
49m2t8 23. 
64r6 24. 
3
22.  w15 (2x)8
25
7| m | t 4
4|m|
 4r 2w 5 16x 4
5
25. 
625s8
4
26. 
216p3
3
q9 27. 
676x4 
y6 28. 
3
27x9
y12 46. 
25x4y6 5x2y3 47. 
36x4z4 6x 2z 2 48. 
169x8y4 13x 4y2
5s 2 6pq 3 26x 2| y 3| 3x 3y 4

49.  50.  51. 

27c9
3 3
29. 144m
 30. 
32x5 31. 
(m   32. 
(2x 
9p12q6 3p6q3 8a3b3 2ab d12 3c3d 4
8n6 5 6 3
y10 4)6 1)3
12m 4| n 3| 2xy 2 | m  4| 2x  1

52. 
(4x  
y)2 4x  y 53. 
(p  q ★ 54. 
x2  4
3
33. 
49a10 34. 
(x  5 35.  36. 
x2  1 
54. x  2 )3 p  q x4
4 3
b16 )8 343d6 0x  25
7| a 5 | b8 (x  5)2 7d 2 | x  5|
37. RADIANT TEMPERATURE Thermal sensors measure an object’s radiant temperature,
which is the amount of energy radiated by the object. The internal temperature of an
4
★ 55. z2  8
z  16 ★ 56.  4a2  
4a  1 ★ 57. 9x2  12x 4
z  4 2a  1
object is called its kinetic temperature. The formula Tr  Tke relates an object’s radiant
temperature Tr to its kinetic temperature Tk. The variable e in the formula is a measure not a real number
of how well the object radiates energy. If an object’s kinetic temperature is 30°C and
e  0.94, what is the object’s radiant temperature to the nearest tenth of a degree?
29.5C
58. Find the principal fifth root of 32. 2
59. What is the third root of 125? 5
38. HERO’S FORMULA Salvatore is buying fertilizer for his triangular garden. He knows
the lengths of all three sides, so he is using Hero’s formula to find the area. Hero’s
formula states that the area of a triangle is  s(s  
a)(s b)(s c), where a, b, and c are
the lengths of the sides of the triangle and s is half the perimeter of the triangle. If the
lengths of the sides of Salvatore’s garden are 15 feet, 17 feet, and 20 feet, what is the
area of the garden? Round your answer to the nearest whole number. 124 ft2 60. SPORTS Refer to the drawing at the right.
Gl NAME
/M G ______________________________________________
Hill 266 DATE ____________
Gl PERIOD
Al _____
b 2
How far does the catcher have to throw a ball 2nd
Reading
5-5 Readingto to Learn
Learn MathematicsELL base
Mathematics, p. 267 from home plate to second base? about 127.28 ft
Roots of Real Numbers
90 ft 90 ft
Pre-Activity How do square roots apply to oceanography?
Read the introduction to Lesson 5-5 at the top of page 245 in your textbook. 61. FISH The relationship between the length and 3rd 1st
pitcher
Suppose the length of a wave is 5 feet. Explain how you would estimate the
speed of the wave to the nearest tenth of a knot using a calculator. (Do not mass of Pacific halibut can be approximated by base base
actually calculate the speed.) Sample answer: Using a calculator,
the equation L  0.46
3
find the positive square root of 5. Multiply this number by 1.34.
Then round the answer to the nearest tenth.
M, where L is the length home
90 ft plate 90 ft
in meters and M is the mass in kilograms.
Reading the Lesson
Use this equation to predict the length of a catcher
1. For each radical below, identify the radicand and the index. 25-kilogram Pacific halibut. about 1.35 m
3
a. 23
 radicand: 23 index: 3

b. 
15x2 radicand: 15x 2 index: 2 248 Chapter 5 Polynomials
c. 343

5
radicand: 343 index: 5

2. Complete the following table. (Do not actually find any of the indicated roots.)

Number of Positive Number of Negative Number of Positive Number of Negative NAME ______________________________________________ DATE ____________ PERIOD _____
Number
Square Roots Square Roots Cube Roots Cube Roots

27 1 1 1 0 Enrichment,
5-5 Enrichment p. 268
16 0 0 0 1
Approximating Square Roots
3. State whether each of the following is true or false.
Consider the following expansion.
a. A negative number has no real fourth roots. true 2 b2
(a  2ba) 2ab
 a2    2
2a 4a
b. 121
 represents both square roots of 121. true b2
 a2  b  2
4a
c. When you take the fifth root of x5, you must take the absolute value of x to identify b2
Think what happens if a is very great in comparison to b. The term 2 is very
the principal fifth root. false 4a
small and can be disregarded in an approximation.
b 2
Helping You Remember ( )
a    a2  b
2a
b
4. What is an easy way to remember that a negative number has no real square roots but a    
a2  b
2a
has one real cube root? Sample answer: The square of a positive or negative
number is positive, so there is no real number whose square is negative. Suppose a number can be expressed as a2  b, a b. Then an approximate value
However, the cube of a negative number is negative, so a negative b b
of the square root is a  . You should also see that a    
a2  b.
number has one real cube root, which is a negative number. 2a 2a

Example Use the formula  

a2  b
b

a   to approximate 101
 and 622
.
2a

248 Chapter 5 Polynomials

 62. SPACE SCIENCE The velocity v required for an object to escape the gravity of
a planet or other body is given by the formula v  
2GM
, where M is the mass
R
About the Exercises…
of the body, R is the radius of the body, and G is Newton’s gravitational constant. Organization by Objective
Use M  5.98
1024 kg, R  6.37
106 m, and G  6.67
1011 N  m2/kg2
to find the escape velocity for Earth. about 11,200 m/s
• Simplify Radicals: 28–59
• Approximate Radicals with
63. CRITICAL THINKING Under what conditions does 
x2  y2  x  y?
a Calculator: 16–27
x  0 and y  0, or y  0 and x  0 Odd/Even Assignments
64. WRITING IN MATH Answer the question that was posed at the beginning of Exercises 16–59 are structured
the lesson. See margin. so that students practice the
Space Science
The escape velocity How do square roots apply to oceanography? same concepts whether they
for the Moon is about Include the following in your answer: are assigned odd or even
2400 m/s. For the Sun, • the values of s for   2, 5, and 10 feet, and
it is about 618,000 m/s. problems.
Source: NASA • an observation of what happens to the value of s as the value of  increases.
Assignment Guide
Standardized ? B
65. Which of the following is closest to 7.32 Basic: 17–53 odd, 59, 61,
Test Practice A 2.6 B 2.7 C 2.8 D 2.9 63–82
66. In the figure, ABC is an equilateral triangle with B D
Average: 17–61 odd, 63–82
sides 9 units long. What is the length of 
BD in units? D 30˚ Advanced: 16–62 even, 63–76
A 3 B 9 (optional: 77–82)
C 92 D 18 A C

Maintain Your Skills

4 Assess
Open-Ended Assessment
Mixed Review Factor completely. If the polynomial is not factorable, write prime. (Lesson 5-4)
Modeling Have students use the
67. 7xy2(y  2xy3  67. 7xy3  14x2y5  28x3y2 68. ab  5a  3b  15 (a  3)(b  5)
4x2) hypotenuse of a right triangle with
69. 2x2  15x  25 (2x  5)(x  5) 70. c3  216 (c  6)(c2  6c  36)
both legs one unit long to demon-
Simplify. (Lesson 5-3) 71. 4x2  x  5  
8 strate and explain that 2 is a
x  2 x4  4x3  4x2  5x
71. (4x3  7x2  3x  2)  (x  2) 72.  x3  x2  x number on the real number line.
x5
73. TRAVEL The matrix at the right shows the costs New
of airline flights between some cities. Write a matrix York LA Getting Ready for
that shows the costs of two tickets for these flights. Lesson 5-6
(Lesson 4-2)
 810
  
2320
1418 2504 
Atlanta
Chicago 709405 1160
1252 PREREQUISITE SKILL Lesson 5-6
presents operations with radical
Solve each system of equations by using either substitution or elimination. expressions. In Example 5 on
(Lesson 3-2)
p. 253, they will encounter the
74. a  4b  6 75. 10x  y  13 76. 3c  7d  1
3a  2b  2 (2, 2) 3x  4y  15 (1, 3) 2c  6d  6 (9, 4) multiplication of two binomials
involving radical expressions.
Getting Ready for PREREQUISITE SKILL Find each product. Exercises 77–82 should be used
the Next Lesson (To review multiplying binomials, see Lesson 5-2.) to determine your students’
77. (x  3)(x  8) x2  11x  24 78. (y  2)(y  5) y 2  3y  10 familiarity with multiplying
79. (a  2)(a  9)  7a  18
a2 80. (a  b)(a  2b) a2  3ab  2b 2 binomials.
81. (x  3y)(x  3y) x 2  9y 2 82. (2w  z)(3w  5z) 6w 2  7wz  5z 2
www.algebra2.com/self_check_quiz Lesson 5-5 Roots of Real Numbers 249 Assessment Options
Quiz (Lessons 5-4 and 5-5) is
available on p. 307 of the Chapter 5
Answer Resource Masters.
64. The speed and length of a wave are Mid-Chapter Test (Lessons 5-1
related by an expression containing a through 5-5) is available on
square root. Answers should include the p. 309 of the Chapter 5 Resource
following. Masters.
• about 1.90 knots, about 3.00 knots,
and 4.24 knots
• As the value of  increases, the value
of s increases.
Lesson 5-5 Roots of Real Numbers 249
 Lesson Radical Expressions
Notes

• Simplify radical expressions.

1 Focus • Add, subtract, multiply, and divide radical expressions.

Vocabulary do radical expressions apply to falling objects?

5-Minute Check • rationalizing the The amount of time t in seconds that it takes for an object to drop d feet is given
Transparency 5-6 Use as a denominator
quiz or a review of Lesson 5-5. • like radical expressions by t   2d
, where g  32 ft/s2 is the acceleration due to gravity. In this lesson,
g
you will learn how to simplify radical expressions like  
• conjugates 2d
.
Mathematical Background notes g
are available for this lesson on
p. 220D.
SIMPLIFY RADICAL EXPRESSIONS You can use the Commutative
Property and the definition of square root to find an equivalent expression for
Building on Prior a product of radicals such as 3  5. Begin by squaring the product.
Knowledge 3  5 2  3  5  3  5
In Lesson 5-5, students simplified  3  3  5  5 Commutative Property of Multiplication
radicals. In this lesson, students  3  5 or 15 Definition of square root
build on the skills they learned Since 3  5 0 and 3  5   15, 3  5 is the principal square root of 15.
2

in that lesson to simplify and

That is, 3  5  15
. This illustrates the following property of radicals.
combine radical expressions.

do radical expressions Product Property of Radicals

apply to falling objects? For any real numbers a and b and any integer n 1,
  a  b
1. if n is even and a and b are both nonnegative, then ab , and
n n n
Ask students:
  a  b
2. if n is odd, then ab .
n n n
• If the value of d in the formula
doubles, will the value of t also
double? no Follow these steps to simplify a square root.
• Is the relationship between Step 1 Factor the radicand into as many squares as possible.
time t and distance d for a
Step 2 Use the Product Property to isolate the perfect squares.
falling object linear or
nonlinear? nonlinear Step 3 Simplify each radical.

Example 1 Square Root of a Product

Simplify 
16p8q7.

16p8q7   42  (p4
)2  (q3
)2  q Factor into squares where possible.

 42  (p  4)2 

(q  3)2 
q Product Property of Radicals

 4p q q
4 3 Simplify.

However, for  16p8q7 to be defined, 16p8q7 must be nonnegative. If that is true,

q must be nonnegative, since it is raised to an odd power. Thus, the absolute value
is unnecessary, and  16p8q7  4p4q3q.

250 Chapter 5 Polynomials

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 5 Resource Masters Teaching Algebra With Manipulatives 5-Minute Check Transparency 5-6
• Study Guide and Intervention, pp. 269–270 Masters, p. 238 Real-World Transparency 5
• Skills Practice, p. 271 Answer Key Transparencies
• Practice, p. 272
• Reading to Learn Mathematics, p. 273 Technology
• Enrichment, p. 274 Interactive Chalkboard
 Look at a radical that involves division to see if there is a quotient property for
49
radicals that is similar to the Product Property. Consider . The radicand is a
9 2 Teach
perfect square, so  
 
49
 
9
7 2 7 7  
49
 or . Notice that    . This suggests the
3 3 3 9 SIMPLIFY RADICAL
following property.
EXPRESSIONS
Quotient Property of Radicals
In-Class Examples Power
Point®
• Words For any real numbers a and b  0, and any integer n 1,

ba  
a , if all roots are defined.
n
n


b
n Teaching Tip When discussing
the Product Property of Radicals,
• Example 

27
 or 3
 9 stress the fact that a and b must
3
both be nonnegative if n is even.
This means that  2 times
You can use the properties of radicals to write expressions in simplified form.  8 may not be written as
 16. This condition is necessary
Simplifying Radical Expressions because  2 and  8 are not
real numbers.
A radical expression is in simplified form when the following conditions are met.
• The index n is as small as possible. 1 Simplify 
25a 4b9.
• The radicand contains no factors (other than 1) that are nth powers of an integer 5a 2b 4b
or polynomial.
• The radicand contains no fractions. 2 Simplify each expression.


• No radicals appear in a denominator.
y 8 y 4x
Study Tip a. 7 
x x4


Rationalizing 3
the Denominator
To eliminate radicals from a denominator or fractions from a radicand, you can
32 

2
6x
use a process called rationalizing the denominator. To rationalize a denominator, b. 
You may want to think 9x 3x
multiply the numerator and denominator by a quantity so that the radicand has an
of rationalizing the
exact root. Study the examples below.
denominator as making Teaching Tip Urge students to
the denominator a verify that each of their final
rational number. Example 2 Simplify Quotients answers is in simplified form by
Simplify each expression. testing it against the four con-
ditions listed in the Concept
 
4
x 5 5
a. 
5 b.  Summary for Simplifying Radical
y 4a
Expressions.
5
xy  


5
4 x 4
5
 
5
Quotient Property Quotient Property
5
 
4a
5
y 5 4a

 5  

2 2 5 5
(x ) 8a4 Rationalize the
  Factor into squares. 
   
4a 4
8a denominator.
5 5
(y2)2  y

 
5
(x ) 2 2 5  8a4
  Product Property   Product Property
 
(y )  y 
4a  8a
2 2 5 4


40a
5 4
x 2
  
(x2)2  x2  Multiply.
y 
5 5
2y 32a

x 2 y Rationalize the 

5
40a4
     
5
32a5  2a
2y y y denominator. 2a


x2  y
  3 y  y  y
y

www.algebra2.com/extra_examples Lesson 5-6 Radical Expressions 251

Differentiated Instruction
Intrapersonal Have students think about irrational numbers, radicals,
and the rules for operations with radicals. Ask them to write down what
puzzles them most about these concepts, including a list of the definitions
and operations about which they feel some confusion. Invite students to
share these concerns with you so that they can be cleared up.

Lesson 5-6 Radical Expressions 251

OPERATIONS WITH OPERATIONS WITH RADICALS You can use the Product and Quotient
Properties to multiply and divide some radicals, respectively.
RADICALS
In-Class Example Power Example 3 Multiply Radicals
Point®
3 3
3 3 Simplify 6 .
9n2  324n
3 Simplify 5
100a2  10a
.
6   6  3  
9n2  324n 
3 3 3
50a 9n2  24n Product Property of Radicals

 18  
23  33
3
 n3 Factor into cubes where possible.
Teaching Tip After discussing
 18      
3 3 3
the information presented in 23 33 n3 Product Property of Radicals

the Algebra Activity, make sure  18  2  3  n or 108n Multiply.

students also understand that
a   b is not equivalent to
Can you add radicals in the same way that you multiply them? In other words, if
a  b. Suggest they use the
values a  16 and b  9 to a  a  
a  a, does a  a  
a  a?
verify this fact.

Adding Radicals
  2.
You can use dot paper to show the sum of two like radicals, such as 2
Model and Analyze
Step 1 First, find a segment Step 2 Extend the segment
of length 2  units by using to twice its length to represent
the Pythagorean Theorem 2  2.
with the dot paper.
a2  b2  c2
12  12  c2 2
2  c2 1
1 2

1 2
1

Make a Conjecture
1. No; 2  2  1. Is 2  2   2  2 or 2? Justify your answer using the geometric
units is the length models above.
of the hypotenuse of 2. Use this method to model other irrational numbers. Do these models
an isosceles right support your conjecture? See students’ work.
triangle whose legs
have length 2 units.
Therefore,
2  2 2. In the activity, you discovered that you cannot add radicals in the same manner as
you multiply them. You add radicals in the same manner as adding monomials.
That is, you can add only the like terms or like radicals.
Study Tip Two radical expressions are called like radical expressions if both the indices and
the radicands are alike. Some examples of like and unlike radical expressions are
Reading Math given below.
Indices is the plural
of index.
3 and 3 are not like expressions.
3
Different indices

5x  and 5 are not like expressions.

4 4
Different radicands

23a  and 53a are like expressions.

4 4
Radicands are 3a; indices are 4.

252 Chapter 5 Polynomials

Algebra Activity
Materials: rectangular dot paper, ruler/straightedge
• Ask students what leg lengths they could use on a right triangle to find a line
whose length is  5. lengths of 1 and 2 units
• Point out that there are other instances where you can perform a multiplication
but not an addition. For example, you can multiply fractions by multiplying the
numerators and denominators separately, but you do not add fractions this way.

252 Chapter 5 Polynomials

Example 4 Add and Subtract Radicals In-Class Examples Power
Point®
  327
Simplify 212   248
.
4 Simplify
  327
212   248

 2
22  3  3
32  3  2
22  22
3 Factor using squares.
345
  580
  420
.
 2
22  3  3
32  3  2
22  
22  3 Product Property
35
 2  2  3  3  3  3  2  2  2  3 22  2, 
32  3 5 Simplify each expression.
 43  93  83 Multiply.
a. (23  35 )(3  3 )
 33 Combine like radicals.
63  6  95  315

b. (42  7)(42  7) 17
Just as you can add and subtract radicals like monomials, you can multiply 2  3
 11  63
radicals using the FOIL method as you do when multiplying binomials. 6 Simplify  . 
4  3
 13

Example 5 Multiply Radicals

a. 35  23 2  3 
F O I L
35  23 2  3   35  2  35  3  23  2  23  3
  3
 65 5  3  43  2
32 Product Property

  315
 65   43  6 2
32  2  3 or 6
b. 53  653  6
53  653  6  53  53  53  6  6  53  6  6 FOIL
 2532  303  303  36 Multiply.
 75  36 25
32  25  3 or 75
 39 Subtract.

Binomials like those in Example 5b, of the form ab  cd and ab  cd
where a, b, c, and d are rational numbers, are called conjugates of each other. The
product of conjugates is always a rational number. You can use conjugates to
rationalize denominators.

Example 6 Use a Conjugate to Rationalize a Denominator


1  3
Simplify  .
5   3

1  3 1  35  3  because 5  3 is the conjugate of 5  
5  3
   Multiply by   3.
5  
3 5  35  3 
5  3

1  5  1  3  3  5  3 2 FOIL

 
5  3
2 2 Difference of squares

    53  3
5  3 Multiply.
25  3

  
8  63
Combine like terms.
22

  
4  33 Divide numerator and denominator by 2.
11

Lesson 5-6 Radical Expressions 253

Unlocking Misconceptions
Radical Expressions When presented with a radical expression such as
11  6 3, some students may persist in trying to add the 11 and the 6.
Help them understand why this cannot be done by comparing this radical
expression 11  6 3 to the expression 11  6x. Stress that the radical
6 3 is a multiplication expression just like 6x. Remind students that the
order of operations requires that multiplication be performed before addi-
tion. Students may find it helpful to rewrite 11  6 3, as 11  6   3.

Lesson 5-6 Radical Expressions 253

 3 Practice/Apply Concept Check 1
1. Determine whether the statement   a is sometimes, always, or never true.
n

 a
n
1 Explain.
1. Sometimes; 
n 
n a 2. OPEN ENDED Write a sum of three radicals that contains two like terms.
a only when 3. Explain why the product of two conjugates is always a rational number.
Study Notebook a  1.
2–3. See margin.
Have students— Guided Practice Simplify.
 
4 7
 157
4. 563 16x5y4 2xyx
5.  14y
4
• add the definitions/examples of GUIDED PRACTICE KEY 6. 
8y 
4y
the vocabulary terms to their Exercises Examples 
625
3
3
7. 215
 421
 8.  
25 9. 
2ab2  
6a3b2
Vocabulary Builder worksheets for 
3
4–5, 14 1 25
6 2 3
Chapter 5. 10. 3  23  43  53  222
  516
11. 3128
4 4 3 3
7–9 3
• add the information from the Key 10, 11 4 
1  5
12 5 12. 3  5 1  3  
13.  2  5

3  5
Concept and Concept Summary 13 6 3  33
  5  15

features to their notebook. Application 14. LAW ENFORCEMENT

• include any other item(s) that they 
7. 2435
A police accident
investigator can use
find helpful in mastering the skills

9. 2a2b23 the formula s  2 5
in this lesson. 4 to estimate the speed s
10. 53  33
 of a car in miles per hour
based on the length  in feet of the skid marks it left. How fast was a car
traveling that left skid marks 120 feet long? about 49 mph

★ indicates increased difficulty

About the Exercises… Practice and Apply

3 3
Organization by Objective Homework Help Simplify. 21. 3xy
2y 22. 2ab2
10a 23. 6y2z7 24. 4mn 3mn2
3 4
 93
15. 243  62
16. 72  32
17. 54  26
18. 96
For See 3 4
• Simplify Radical Exercises Examples
Expressions: 15–30 15–26 1 19.   20. 
50x4 5x22  21. 
16y3 2y 2
3
18x2y3
3
22. 
40a3b4
• Operations with Radicals: 27–30 2

31–48
31–34
35–38
3
4
23. 3
3
56y6z3
3
24. 2
24m4n5
3
25. 
4 1
c d
81
5 4
26. 
5 1
32
w z
6 7

39–42 5 4
6  a2b 4r8 2r4t
Odd/Even Assignments 34 23  
54 a 4
43–48 6 27.
3
 28.
4
 29. 
 2 3 30. 9 
Exercises 15–48 are structured 2 3 b b t t5
so that students practice the Extra Practice 31. 312   367
 221 
32. 3   6030
24 520  
See page 838.
same concepts whether they 4
6
are assigned odd or even 25. 1cdc  divided by 26
33. What is 39 ? 
3 2
problems. 5 34. Divide 14 by  
35.  10
26. 1wz 
wz2 5
2
Assignment Guide Simplify. 37. 7
3  22 38. 45  236 39. 25  52  56  23
Basic: 15–45 odd, 49–51, 55–82   48
35. 12  33
  27   72
36. 98  52
  32
37. 3  72
  128
  108
   24
38. 520   180
  754

Average: 15–49 odd, 50–53,
55–82

40. 6  36 39. 5  6 5  2  40. 3  7 2  6

27  42
Advanced: 16–48 even, 50–74 41. 11 
  2 2 13  222 
 2 8  215
42. 3  5
(optional: 75–82) 7
43.   28  7
3 6
44.   5
6  3  2  3 1  3
2 45.  

4  3 13 5  3 22 1  3 2
All: Practice Quiz 2 (1–10)
2  2 12  7 x1 x2  1 ★ 48. x1
46.   2 ★ 47. 
 x  1
5  2 23 x2  1 x1 x  1
254 Chapter 5 Polynomials

Answers
2. Sample answer: 2  3  2 50. The square root of a difference is not the difference of the square roots.
3. The product of two conjugates 56. The formula for the time it takes an object to fall a certain distance can be written in
yields a difference of two squares. various forms involving radicals. Answers should include the following.
Each square produces a rational 2d
 . Multiply by g
• By the Quotient Property of Radicals, t    to rationalize the
number and the difference of two g g
rational numbers is a rational 2dg

denominator. The result is  .
g
number.
• about 1.12 s
254 Chapter 5 Polynomials
 49. GEOMETRY Find the perimeter and area of the 3  6 2 yd Study
NAME ______________________________________________ DATE

Guide andIntervention
Intervention,
____________ PERIOD _____

5-6 Study Guide and

rectangle. 6  162
 yd, 24  62 yd2 p. 269
8 yd Radical(shown)
Expressions and p. 270
Simplify Radical Expressions

Lesson 5-6
For any real numbers a and b, and any integer n 1:
n n n
Product Property of Radicals 1. if n is even and a and b are both nonnegative, then ab
  a  b.
n n n
2. if n is odd, then ab
  a  b.
AMUSEMENT PARKS For Exercises 50 and 51, use the following information. To simplify a square root, follow these steps:
The velocity v in feet per second of a roller coaster at the bottom of a hill is related to 1. Factor the radicand into as many squares as possible.
2. Use the Product Property to isolate the perfect squares.
the vertical drop h in feet and the velocity v0 in feet per second of the coaster at the 3. Simplify each radical.

For any real numbers a and b  0, and any integer n 1,

top of the hill by the formula v0   .

v2  64h ab  ab , if all roots are defined.
Quotient Property of Radicals n
n
n

To eliminate radicals from a denominator or fractions from a radicand, multiply the

50. Explain why v0  v  8h is not equivalent to the given formula. See margin.
numerator and denominator by a quantity so that the radicand has an exact root.


8x3
51. What velocity must a coaster have at the top of a 225-foot hill to achieve a Example 1 Simplify 16a
5
3
b7 . Example 2 Simplify .
45y 5

16a


3 5b7  
(2)3 
 2  a
3 3  a2 
 (b2) 3
b
velocity of 120 feet per second at the bottom? 0 ft/s  2ab22a
3
 2b 
8x3
45y5



8x3
45y5
Quotient Property

(2x)
 2 2x
  Factor into squares.
 (3y2)2
Online Research Data Update What are the values of v and h (2x)
 
2  2x
 5y

Product Property
(3y
 2)2  5y
for some of the world’s highest and fastest roller coasters? Visit
Amusement www.algebra2.com/data_update to learn more.
2| x|2x
 
3y25y


Simplify.

Parks  
2
2| x|2x

3y 5y

 5y

5y

Rationalize the
denominator.

2| x|10xy

Attendance at the top SPORTS For Exercises 52 and 53, use the following information.  
3 15y
Simplify.

50 theme parks in North

A ball that is hit or thrown horizontally with a velocity of v meters per second will
America increased to Exercises
175.1 million in 2000.
Source: Amusement Business
travel a distance of d meters before hitting the ground, where d  v  and h is the
4.9  h Simplify.

 156
1. 554  2. 
4
 2a 2|b 5| 2a
32a9b20 
4
75x4y7 5x 2y 3 5y
3.  
height in meters from which the ball is hit or thrown.
    
3
pq 
4.9h 36 65 a6b3 |a 3 |b2b
 5p
p5q3
2

52. Use the properties of radicals to rewrite the formula. d  v 

4.   5.   6. 3

125 25 98 14 10 40

4.9
53. How far will a ball that is hit horizontally with a velocity of 45 meters per Gl NAME
/M G ______________________________________________
Hill 269 DATE ____________
GlPERIOD
Al _____
b 2

Skills
5-6 Practice,
Practice p. 271 and
second at a height of 0.8 meter above the ground travel before hitting the Practice,
(Average)
p. 272 (shown)
Radical Expressions
ground? about 18.18 m Simplify.
3 3
 615
  62
  42

3 3
1. 540 2. 432 3. 128

54. AUTOMOTIVE ENGINEERING An automotive engineer is trying to design  35

4. 405
4

4 3
5. 500
0  10 5

3
6. 121
5  35
5

5

a safer car. The maximum force a road can exert on the tires of the car being
5t 2 
3 4
2gk 2 
3
redesigned is 2000 pounds. What is the maximum velocity v in ft/s at which 7. 
3
125t6w2 w2 8. 
4

48v8z13 2v 2z 33z
 9. 
3
8g3k8 k2

   9
 Fr 45x3y8 3xy 45x
10.   11

11
 3 216

3

this car can safely round a turn of radius 320 feet? Use the formula v 
11. 12.
c, 39 24

100
 161 c d 2d    
4

13.
1
 c4d 7  2 3 14. 3a a
9a5
  2
15.

72a
8
4 
128 64b4 2 3a 3
9a
where Fc is the force the road exerts on the car and r is the radius of the turn. 8b

80 ft/s or about 55 mph 16. (315

 )(445
1803

) 17. (224
 )(718
1683

) 18. 810
  240
410
  250

  415

55. CRITICAL THINKING Under what conditions is the equation 
x3y2  xyx
19. 620
  85
  545
 20. 848
  675
  780
 21. (32
  23
 )2
true? x and y are nonnegative. 55
 23
  285
 30  126


22. (3  7
 )2 23. (5
  6
 )(5
  2
) 24. (2
  10
 )(2
  10
)
56. WRITING IN MATH Answer the question that was posed at the beginning of 16  67
 5  10
  30
  23
 8

the lesson. See margin. 25. (1  6

 )(5  7
) 26. (3
  47
 )2 27. (108
  63
 )2
5  7
  56
  42
 115  821
 0
How do radical expressions apply to falling objects?
3 6 17  3

5  3

28.  15
  23
 29.  62
6 30.  
Include the following in your answer: 5
2 2
1 4  3
 13

 8  52
3  2  3  6 3  x 6  5x
x
31.   32.  27  116
 33.  
• an explanation of how you can use the properties in this lesson to rewrite the 2  2
 2 5  24
 2  x 4x

formula t  
2d
, and
g
34. BRAKING The formula s  25  estimates the speed s in miles per hour of a car when
it leaves skid marks  feet long. Use the formula to write a simplified expression for s if
  85. Then evaluate s to the nearest mile per hour. 1017 ; 41 mi/h
35. PYTHAGOREAN THEOREM The measures of the legs of a right triangle can be
• the amount of time a 5-foot tall student has to get out of the way after a represented by the expressions 6x2y and 9x2y. Use the Pythagorean Theorem to find a
simplified expression for the measure of the hypotenuse. 3x 2 | y | 13

balloon is dropped from a window 25 feet above.
Gl NAME
/M G ______________________________________________
Hill 272 DATE ____________
Gl PERIOD
Al _____
b 2
Reading
5-6 Readingto
to Learn
Learn Mathematics
Standardized  is equivalent to which of the following? B
57. The expression 180 Mathematics, p. 273 ELL
Radical Expressions
Test Practice A 56 B 65 C 
310 D 365 Pre-Activity How do radical expressions apply to falling objects?

Lesson 5-6
Read the introduction to Lesson 5-6 at the top of page 250 in your textbook.
Describe how you could use the formula given in your textbook and a
calculator to find the time, to the nearest tenth of a second, that it would
58. Which of the following is not a length of a side take for the water balloons to drop 22 feet. (Do not actually calculate the
time.) Sample answer: Multiply 22 by 2 (giving 44) and divide

of the triangle? D 2
by 32. Use the calculator to find the square root of the result.
Round this square root to the nearest tenth.

A 8 B 22 Reading the Lesson

6
4 2 4  2
1. Complete the conditions that must be met for a radical expression to be in simplified form.
C D • The index n is as small as possible.

• The radicand contains no factors (other than 1) that are nth

www.algebra2.com/self_check_quiz Lesson 5-6 Radical Expressions 255 powers of a(n) integer or polynomial.

• The radicand contains no fractions .

• No radicals appear in the denominator .

2. a. What are conjugates of radical expressions used for? to rationalize binomial

NAME ______________________________________________ DATE ____________ PERIOD _____ denominators
1  2

Enrichment,
5-6 Enrichment p. 274 b. How would you use a conjugate to simplify the radical expression  ?
Multiply numerator and denominator by 3  2
.
3  2


c. In order to simplify the radical expression in part b, two multiplications are

Special Products with Radicals necessary. The multiplication in the numerator would be done by the FOIL
2 method, and the multiplication in the denominator would be done by finding the
Notice that (3
 )(3
 )  3, or (3
 )  3.
2
In general, (x )  x when x  0. difference of two squares .

Also, notice that (9

)(4
)  36
.
Helping You Remember
In general, (x )(y )  xy
 when x and y are not negative.
3. One way to remember something is to explain it to another person. When rationalizing the
You can use these ideas to find the special products below.
1
(a  b )(a  b )  (a)2  (b )2  a  b denominator in the expression 
3 , many students think they should multiply numerator
2

(a  b )2  (a )2  2ab
  (b  )2  a  2abb
and denominator by 
3
2

. How would you explain to a classmate why this is incorrect
3
(a  b )2  (a )2  2ab
  (b  )2  a  2abb 2

and what he should do instead. Sample answer: Because you are working with
cube roots, not square roots, you need to make the radicand in the
Example 1 Find the product: (2  )(2
  5  ).
  5 denominator a perfect cube, not a perfect square. Multiply numerator and
3
(2  5 )(2  5 )  (2)2  (5 )2  2  5  3 4 3
denominator by 
3 to make the denominator 8
, which equals 2.
4

Example 2 Evaluate (2 ) .
  8 2

(2  8)2 (2)2  228  (8)2

Lesson 5-6 Radical Expressions 255

 Maintain Your Skills
4 Assess Mixed Review Simplify. (Lesson 5-5)
59.   12z4 60. 
216a3b9 6ab3 61. 
(y  2
)2 y  2
3
144z8
Open-Ended Assessment
Speaking Ask students to Simplify. Assume that no denominator is equal to 0. (Lesson 5-4)
describe how combining radicals x2  5x  14 x  7 x2  3x  4 x  1
62.    63.   
is the same as combining x2  6x  8 x  4 x2  16 x4
expressions with variables, and Perform the indicated operations. (Lesson 4-2)
how it differs from working with 3 4 5 2 4
  
0
variables. 64. 2 8  7 7 9 15 65. 30 3
2 

2
5
1
2 51 4
4 
0 1 3 6 3 5

66. Find the maximum and minimum values of the function f(x, y)  2x  3y for the
Intervention region with vertices at (2, 4), (1, 3), (3, 3), and (2, 5). (Lesson 3-4) 16, 15
New Students will 67. State whether the system of equations shown at y
need to simplify the right is consistent and independent, consistent and
expressions in- dependent, or inconsistent. (Lesson 3-1)
volving radicals consistent and independent O
x
in much of their further work
in algebra. Take time to help
students uncover and correct
their misconceptions by ana-
lyzing the errors they make. 68. BUSINESS The amount that a mail-order company charges for shipping and
handling is given by the function c(x)  3  0.15x, where x is the weight in
pounds. Find the charge for an 8-pound order. (Lesson 2-2) $4.20

Getting Ready for Solve. (Lessons 1-3, 1-4, and 1-5)

69. 2x  7  3 5 70. 5x  6  4 2
Lesson 5-7 7
71. |x  1|  3 2, 4 72. |3x  2|  5 , 1
BASIC SKILL Lesson 5-7 presents 3
73. 2x  4 8 {xx 6} 74. x  3  4 {xx  7}
working with rational exponents.
This often involves adding, Getting Ready for BASIC SKILL Evaluate each expression.
subtracting, or multiplying 5 3 13
75. 2 1 76. 3 1
1 1 1 1 1
the Next Lesson 77.    
6
78.    
4 12
fractions. Exercises 75–82 should 8 4 6 2 2 3 3
be used to determine your 1 5 13 5 1 19 5 1 3 1 2 5
79.     80.     81.     82.    
8 12 24 6 5 30 8 4 8 4 3 12
students’ familiarity with
rational numbers.

Assessment Options
Practice Quiz 2 The quiz pro-
P ractice Quiz 2 Lessons 5-4 through 5-6
vides students with a brief review Factor completely. If the polynomial is not factorable, write prime. (Lesson 5-4)
of the concepts and skills in 1. 3x3y  x2y2  x2y x2y(3x  y  1) 2. 3x2  2x  2 prime
Lessons 5-4 through 5-6. Lesson 3. ax2  6ax  9a a(x  3)2 4. 8r3  64s6 8(r  2s2)(r2  2rs2  4s4)
numbers are given to the right of
exercises or instruction lines so Simplify. (Lessons 5-5 and 5-6)
5.  6. 
64a6
b9 4a2b3 7. 
4n2 
12n 9 2n  3
3
students can review concepts not 36x2y6 6x y3
x2y 5  2 8  32
yet mastered.
 
4
8.
x
 
y 3
y 2
9. 3  7 2  7  1  7 10.  
2  2 2

256 Chapter 5 Polynomials

256 Chapter 5 Polynomials

 Rational Exponents Lesson
Notes

• Write expressions with rational exponents in radical form, and vice versa.
• Simplify expressions in exponential or radical form.

do rational exponents apply to astronomy?

1 Focus
Astronomers refer to the space around a planet 5-Minute Check
where the planet’s gravity is stronger than the Transparency 5-7 Use as a
Sun’s as the sphere of influence of the planet. The quiz or review of Lesson 5-6.
r
radius r of the sphere of influence is given by
2
Mp 5
the formula r  D , where Mp is the mass D Mathematical Background notes
MS are available for this lesson on
of the planet, MS is the mass of the Sun, and D is p. 220D.
the distance between the planet and the Sun.
do rational exponents
apply to astronomy?
Ask students:
• Is the p in Mp an exponent? Is it a
RATIONAL EXPONENTS AND RADICALS You know that squaring a variable? No, it is neither an expo-
number and taking the square root of a number are inverse operations. But how
would you evaluate an expression that contains a fractional exponent such as the nent nor a variable; it is a subscript.
one above? You can investigate such an expression by assuming that fractional • Would you expect the radius of
exponents behave as integral exponents.
the sphere of influence for one
2
b2  b2  b2 of the larger planets in our solar
1 1 1
Write the square as multiplication.
1  12 system to be greater than the
b 2
Add the exponents.
radius of the sphere of influence
 b1 or b Simplify. for Earth? Use the formula to
1
Thus, b is a number whose square equals b. So it makes sense to define b 2  b.
2
1
justify your answer. Yes; for the
planets larger than Earth, the value
1 of Mp would be greater than the
1
bn value of Mp for Earth while the value
, except
For any real number b and for any positive integer n, b  b
n
• Words n
of MS is the same. So the value of
when b  0 and n is even.
Mp
1
 or 2
• Example 8 3  8
3
the ratio  is greater for the
MS
larger planets.

Example 1 Radical Form

Write each expression in radical form.
1
a. a 4
1
a 4  a Definition of bn
4 1

1
b. x 5
1
x 5  x 1
5
Definition of b n

Lesson 5-7 Rational Exponents 257

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 5 Resource Masters Graphing Calculator and 5-Minute Check Transparency 5-7
• Study Guide and Intervention, pp. 275–276 Spreadsheet Masters, p. 36 Answer Key Transparencies
• Skills Practice, p. 277
• Practice, p. 278 Technology
• Reading to Learn Mathematics, p. 279 Interactive Chalkboard
• Enrichment, p. 280
• Assessment, p. 308

Lesson x-x Lesson Title 257

 Example 2 Exponential Form
2 Teach Write each radical using rational exponents.

a. y
3

RATIONAL EXPONENTS y  y 3

3 1

Definition of b n
1

AND RADICALS
b. c
8

In-Class Examples Power

1
Point® c  c 8
8  1
Definition of b n

1 Write each expression in

Many expressions with fractional exponents can be evaluated using the definition
radical form. 1
1
 6 Study Tip of b n or the properties of powers.
a. a a
6
Negative Base
1

Example 3 Evaluate Expressions with Rational Exponents
b. m m
 2 Suppose the base of a
monomial is negative Evaluate each expression.
such as (9)2 or (9)3. 1

2 Write each radical using The expression is
a. 16 4

rational exponents. undefined if the exponent Method 1 Method 2

1
is even because there is 1 1 1
5  
1
16   bn  
1 
 (24)


a. b b 16 16  24
4 4 4
5 no number that, when n
1
 b
1
multiplied an even 16 4


b. w
 w2 number of times, results
 24 4 
1
1
 1 

16 4  16

4
Power of a Power
in a negative number. 4
16 
However, the expression
3 Evaluate each expression. is defined for an odd  4
1
4 16  24  21 Multiply exponents.
1
2 1 exponent.  2
a. 49  1 1
7   Simplify.   21  1
2

2 2 21
b. 32 4 5
3
b. 243 5

Teaching Tip If students are Method 1 Method 2

having difficulty remembering
243  2433 
3 1 3 3
which part of the fractional 5 5
Factor. 243 5  (35) 5 243  35

 35 5 
1 3
exponent is the index, suggest  (2433) 5
Power of a Power Power of a Power
that they recall the basic
 
5 1
2433 
b  b  33
5
1

5
Multiply exponents.
definition b 2   b.

 (3
5
5)3 243  35  27 33  3  3  3

 
35  35
5
 35 Expand the cube.

 3  3  3 or 27 Find the fifth root.

1
In Example 3b, Method 1 uses a combination of the definition of b n and the
properties of powers. This example suggests the following general definition of
rational exponents.

Rational Exponents
• Words For any nonzero real number b, and any integers m and n, with n 1,
m
bm  b
b n     , except when b  0 and n is even.
n n m

2
82  8
• Example 8    2 or 4 3
3 3

m
In general, we define b n as b n  or (bm) n . Now apply the definition of
m 1 1

m
b n to b n  and (bm) n .
1 1 1

m m
bn  
1 1
b (bm) n  
n n
bm
258 Chapter 5 Polynomials

Unlocking Misconceptions
• Exponents Students may be confused because they are not perceiv-
ing and reading the exponent in a way that distinguishes it from a coef-
ficient or multiplier. Ask them to practice reading the exponent correctly,
for example, reading x3 as “x to the third power” or as “x cubed.”.
• Radicals Ask students to practice reading radical expressions correctly,
for example, reading  y3 as “the square root of y cubed.”

258 Chapter 5 Polynomials

 Example 4 Rational Exponent with Numerator Other Than 1 In-Class Example Power
8

Point®
WEIGHT LIFTING The formula M  512  146,230B can be used to estimate 5

the maximum total mass that a weight lifter of mass B kilograms can lift in two Teaching Tip Be sure students
lifts, the snatch and the clean and jerk, combined. notice that the fractional expo-
a. According to the formula, what is the maximum amount that 2000 Olympic nent is negative in the formula
champion Xugang Zhan of China can lift if he weighs 72 kilograms? for the maximum total mass M.
8

M  512  146,230B 5
Original formula
4 WEIGHT LIFTING Use the
8

 512  146,230(72) 5
B  72
formula given in Example 4.
 356 kg Use a calculator.
a. U.S. weightlifter Oscar
The formula predicts that he can lift at most 356 kilograms. Chaplin III competed in the
b. Xugang Zhan’s winning total in the 2000 Olympics was 367.50 kg. Compare same weight class as Xugang
this to the value predicted by the formula. Zhan, finishing in 7th place.
The formula prediction is close to the actual weight, but slightly lower. According to the formula,
Weight Lifting what is the maximum that
With origins in both the Chaplin can lift if he weighs
ancient Egyptian and Greek
societies, weightlifting was
77 kilograms? Source: cnnsi.com
among the sports on the SIMPLIFY EXPRESSIONS All of the properties of powers you learned in The formula predicts that he can
program of the first lift at most 372 kilograms.
Lesson 5 -1 apply to rational exponents. When simplifying expressions containing
Modern Olympic Games, in
rational exponents, leave the exponent in rational form rather than writing the
1896, in Athens, Greece.
expression as a radical. To simplify such an expression, you must write the
b. Oscar Chaplin’s total in the
Source: International
Weightlifting Association expression with all positive exponents. Furthermore, any exponents in the 2000 Olympics was 335 kg.
denominator of a fraction must be positive integers. So, it may be necessary to Compare this to the value
rationalize a denominator. predicted by the formula. The
formula prediction is somewhat
Example 5 Simplify Expressions with Rational Exponents higher than his actual total.
Simplify each expression.
1 7
a. x 5  x 5
1 7
x5  x5  x5  5
1 7
Multiply powers. SIMPLIFY EXPRESSIONS
8
x 5
Add exponents. In-Class Example Power
Point®
3

b. y 4
5 Simplify each expression.
3
 1 1 1 4 5
y 4
 3 bn     
y4
bn a. y 7  y 7 y 7
1 1
 1
 
1 y4 y4 3
2
x3
 3  1

Why use 14 ? b. x 
y4 y4 y x
TEACHING TIP
1

Tell students that if they y4 3 1 3  1
are simplifying an  4 y4  y4  y4 4

expression that was y4

originally written with 1

radicals, they should y4 4
 y 4  y1 or y
write the answer with y
radicals. If the
expression was originally
written with rational
exponents, they should
write the answer with When simplifying a radical expression, always use the smallest index possible.
rational exponents. Using rational exponents makes this process easier, but the answer should be
written in radical form.
www.algebra2.com/extra_examples Lesson 5-7 Rational Exponents 259

Differentiated Instruction
Auditory/Musical Have students name and demonstrate on a
keyboard, guitar, or other instrument, the sounds of the notes described
in Exercises 67 and 68. Other students can work to associate the sound
of the note with the number of vibrations per second given by the
formula.

Lesson 5-7 Rational Exponents 259

In-Class Example Power Example 6 Simplify Radical Expressions
Point®
Simplify each expression.
6 Simplify each expression. 81
8

a. 
3
6
6

16 1
 3
a. 
3 2 3 or 2 
8
 
81
1

81 8
2  1 Rational exponents
3
6 
3 6

6 3
b. 
4x4 
1
2x 2 34 8
  1


81  34

1 1 36
 
y  1 y  2y  1
2 2
32
1


c.     Power of a Power
y1
1
1 
 36
y2  1 1 1
  
 32 6
Quotient of Powers
1
 3 or 3 Simplify.
3
3

3 Practice/Apply b. 
4
9z2
1

4
9z2  (9z2) 4 Rational exponents
1

 (32  z2) 4 9  32

 32   z2 
1 1

Study Notebook 1

4

1
4
Power of a Power

3 z 2 2
Multiply.
Have students—
 3  z 32  3, z2  z
1 1

• add the definitions/examples of

the vocabulary terms to their 
 3z Simplify.
1

Vocabulary Builder worksheets for m 1 2
c. 1


Chapter 5. m2  1
1 1 1
  
• add the information listed in the m2  1
1
m2  1 m2  1
1 1
1
m 2  1 is the conjugate of m 2  1.
1

  
Concept Summary below Example 6 m 1
2
m2  1 m2  1
1

m  2m  1 2
to their notebook.   Multiply.
m1
• include any other item(s) that they
find helpful in mastering the skills
in this lesson. Expressions with Rational Exponents
An expression with rational exponents is simplified when all of the following
conditions are met.
• It has no negative exponents.
• It has no fractional exponents in the denominator.

About the Exercises… • It is not a complex fraction.

• The index of any remaining radical is the least number possible.
Organization by Objective
• Rational Exponents and
Radicals: 21–40
• Simplify Expressions:
41–64
1
Odd/Even Assignments Concept Check 1. OPEN ENDED Determine a value of b for which b 6 is an integer.
Exercises 21–66 are structured 1. Sample answer: 64 1
2. Explain why (16) 2 is not a real number. See margin.
so that students practice the
bm  b  . See margin.
3. Explain why 
n n m
same concepts whether they
are assigned odd or even 260 Chapter 5 Polynomials
problems.
Assignment Guide Answers
Basic: 21–61 odd, 65, 69–84 n 1


Average: 21–65 odd, 69–84 2. In radical form, the 3. In exponential form 

bm is equal to (b m) n . By the
expression would be 16
, 1

m

m

Advanced: 22–66 even, 67, 68, Power of a Power Property, (b m) n  b n . But, b n is also
which is not a real number m
( )
1

70–80 (optional: 81–84) because the index is even equal to b n by the Power of a Power Property.
and the radicand is negative. n m
This last expression is equal to (b ) . Thus,
n n m
bm  (b ) .


260 Chapter 5 Polynomials

 Guided Practice Write each expression in radical form. Study
5-7 Guide
Study
NAME ______________________________________________ DATE

andIntervention
Guide and Intervention,
____________ PERIOD _____

7 5. x  or x 
1 3 2
3 2 3
GUIDED PRACTICE KEY 4. 7 3 3
x2 p. 275 (shown)
Rational Exponents and p. 276
Rational Exponents and Radicals
Exercises Examples
Write each radical using rational exponents. Definition of b n
1
 For any real number b and any positive integer n,
1

b n  b
n
, except when b  0 and n is even.
1 1 5 7
4, 5 1    
 26
6. 26 7.  6 x y
4 3
4
6x5y7 3 3 3
Definition of
m

bn
For any nonzero real number b, and any integers m and n, with n 1,
m

6, 7 2 b n  
bm  (b
n m
 ) , except when b  0 and n is even.
n

8–11 3 Example 1 1
 Example 2 8
 125 
1


Lesson 5-7
Write 28 in radical form.
2 Evaluate  3 .
12–17 5 Evaluate each expression. Notice that 28 0.
1
Notice that 8  0, 125  0, and 3 is odd.

18, 19 6 1 1
 1 2 54 2

28  28
2  1
8
3


8

125 


8. 125 3 5 9. 81 4
10. 27 3 9 11.   
22  7
3 

3
125

20 4 3 3
  
22  7
 2

9 2
 27

5
2

5
Simplify each expression. 5 2

 Exercises
2

1
 1
1 2
x 6 3 1  z 3
a2 b 3 2
12. a 3  a 4 a 12 13.  x 14.  15.    a2b3 Write each expression in radical form.

x
1

6
1
 2z 1

1

b3 a2
1

1

3


1
3
2z 1. 11 7 2. 15 3 3. 300 2
 2 1
z(x  2y) 2 
7 3
 
27 11
 15
 3003
4
mn 3 3
 3
1 1
18. 
  6
17.  16. (mn2) 3
 17. z(x  2y) 2
27x3 3x 19. 
x  2y mn 4
3 Write each radical using rational exponents.

5.  6. 
3 4
4. 47
 3a5b2 162p5
1 1 5 2 1 5
     
47 2 33a3b3 3  24  p4
Application 20. ECONOMICS When inflation causes the price of an item to increase, the new
cost C and the original cost c are related by the formula C  c(1  r)n, where r is Evaluate each expression.
2 2
1
1
 5 

the rate of inflation per year as a decimal and n is the number of years. What 7. 27 3 8. 
25

9. (0.0004) 2
1
would be the price of a $4.99 item after six months of 5% inflation? $5.11 9 
10
0.02

★ indicates increased difficulty 2


10. 8 3  4 2
3
 144 2
11.  1
 

3
1
16 2
12.  1

 
1

27 (0.25) 2

Practice and Apply 32

1

4
1

2

Gl NAME
/M G ______________________________________________
Hill 275 DATE ____________
GlPERIOD
Al _____
b 2

Homework Help Write each expression in radical form. Skills

5-7 Practice,
Practice (Average)
p. 277 and
Practice, p. 278 (shown)
6 4 24. (x2) x2
For See 1 5 1 3 2 4 3
Exercises Examples 21. 6 5
22. 4 3
23. c 5
x2 3 Rational Exponents

c2 or c  2
5 5 Write each expression in radical form.
1 2 4 2
   
21–24 1 1. 5 3 2. 6 5 3. m 7 4. (n3) 5

62 or (6
) m4 or (m
)
3

5 5

2 7 4 7 5
25–28 2 Write each radical using rational exponents. 5
 n n


29–40 3 1 1 1 1 2 1

 23 2
25. 23  62 3
26. 62 27.  28. 
3 4 3
41–52,
16z2 2z 2 5x2y 5 3 x 3 y 3 Write each radical using rational exponents.

5 5. 79
 6. 153

4
7. 
3
27m6n4 8. 5
2a10b
64–66 1
 1 4

1

1

5  2 2 |a 5 | b 2

79 2 153 4 3m 2n 3
53–63 6
Evaluate each expression.
Evaluate each expression.

Extra Practice 1 1 1 1


3
 1 1
5
1
1 3
5
1
 

 
29. 16 4 2 30. 216 3 6 31. 25 2
32. 81 4 9. 81 4 3 10. 1024
4
11. 8
32
See page 838. 5 3 27
2
 1 3
 1 1
  3
5

4
3

1 3
2

1 1

4



12. 256 14. 27 3  27 3 243
 35. 81 2  81 2 81 36. 8 2  8 2 4096
13. (64)
33. (27) 3
34. (32) 5 64 16
9 8 25
2

16
 64 
1 1
5
1 1
 125
216 
2  
64 
3
  15.  
 16.   17. 25
8 1 2 16 2 4 8 3 1
2 3

37.  3  38. 

3 3

1  36 2
 49 4
5
27 39.  40.  343 3

27 3 243
92 3
1

64 3
1
 2 Simplify each expression. 1
4 

   y2
4 3 3 13 1 4 1
    3 5 2
18. g 7  g 7 g 19. s 4  s 4 s 4 20. u u 15 21. y 
y
Simplify each expression. 3
2
 3
 1

2

11
 1

2z  2z 2
1

5 b5 q5 t 12
t3 2z 
2
 23. 2 q 5 24.  

43. b 3  5 b 5 5
1 3 1 22. b 25. 
5 7 3 9    b z1
1 3
 4 5 1

41. y  y y4 42. x  x x3
 
3 3 4 4 q5 5t 2  t z2  1

1 
10
85 22
26.     
27. 12 123
5 4
28. 6
  36

4 a3b
29.    a
1 
2  x6 3b


44. a 3 
1 w5
4 1 10 3b
 6
a 9
45. w

5
 46. x

6  1212
 36


w 1
x 5 30. ELECTRICITY The amount of current in amperes I that an appliance uses can be
3
 1 2
 1  
t 4 t 4 r 3 r 2 a 2 a 12
 RP 
1



calculated using the formula I   2 , where P is the power in watts and R is the
47.  48.  49.  resistance in ohms. How much current does an appliance use if P  500 watts and
6a 3  a 4 6a
1 1
1

1
  
R  10 ohms? Round your answer to the nearest tenth. 7.1 amps
t2 15
r6 3 1
3 

y2  2y 2 
1 
x  3x 2  2
 1 1


2c 8 2c 1 6 y2 x 2  2 
31. BUSINESS A company that produces DVDs uses the formula C  88n 3  330 to
50.   51.   52.  calculate the cost C in dollars of producing n DVDs per day. What is the company’s cost
1

c c
1
 c 1

y2  2 y4 1
 x1 to produce 150 DVDs per day? Round your answer to the nearest dollar. $798
16 4 2
x 1 NAME ______________________________________________ DATE ____________
Gl /M G Hill 278 Gl PERIOD
Al _____
b 2
Reading
Readingto
to Learn
5 3 
172 1717
6

53. 25 
54. 27   
55. 17
4 6 3 5-7 Learn Mathematics
Mathematics, p. 279 ELL
Rational Exponents

53 5  81a4b8 b

6 4 3 Pre-Activity
56. 5   57.  58. 
3 8 6 How do rational exponents apply to astronomy?
55 25x4y4 5x2y2 9a2b Read the introduction to Lesson 5-7 at the top of page 257 in your textbook.
2
The formula in the introduction contains the exponent  . What do you think
xy xyz
3 5

ab
2
c2 it might mean to raise a number to the  power?

59.  
ab

8 2
3 5
60.   61. Sample answer: Take the fifth root of the number and square it.
z c
3
z c
Reading the Lesson

Lesson 5-7
1 1 5 1 4
    
8 9 x x z 1 2

  6
3 6 4
★ 63. ★ 64.
3 3 3
x  x 3z 3
3 1. Complete the following definitions of rational exponents.
62. 36   • For any real number b and for any positive integer n, b n 
1

b

n

3  2
2 2 except
 
x3  z3 when b 0 and n is even .

www.algebra2.com/self_check_quiz 5
26 Lesson 5-7 Rational Exponents 261
• For any nonzero real number b, and any integers m and n, with n

bn 
m


bm
n

n
( )
b
m
, except when b 0
1
and
,

n is even .

2. Complete the conditions that must be met in order for an expression with rational
exponents to be simplified.

NAME ______________________________________________ DATE ____________ PERIOD _____ • It has no negative exponents.

fractional denominator .
Enrichment,
5-7 Enrichment p. 280 • It has no

• It is not a complex
exponents in the

fraction.

• The index of any remaining radical is the least

Lesser-Known Geometric Formulas number possible.
Many geometric formulas involve radical expressions.
3. Margarita and Pierre were working together on their algebra homework. One exercise
4

Make a drawing to illustrate each of the formulas given on this page. asked them to evaluate the expression 27 3 . Margarita thought that they should raise
Then evaluate the formula for the given value of the variable. Round 27 to the fourth power first and then take the cube root of the result. Pierre thought that
answers to the nearest hundredth. they should take the cube root of 27 first and then raise the result to the fourth power.
1. The area of an isosceles triangle. Two 2. The area of an equilateral triangle with Whose method is correct? Both methods are correct.
sides have length a; the other side has a side of length a. Find A when a  8.
length c. Find A when a  6 and c  7. a2
Helping You Remember
A  3

c 4
A  
4a2 
c2
4 a a A  27.71 a a 4. Some students have trouble remembering which part of the fraction in a rational
A  17.06 exponent gives the power and which part gives the root. How can your knowledge of
c integer exponents help you to keep this straight? Sample answer: An integer3

a exponent can be written as a rational exponent. For example, 23  2 1 .
You know that this means that 2 is raised to the third power, so the
numerator must give the power, and, therefore, the denominator must
3. The area of a regular pentagon with a 4. The area of a regular hexagon with a
give the root.
side of length a. Find A when a  4. side of length a. Find A when a  9.
a2 3a2 a
25  
A   105
 a a A  3

4 2

Lesson 5-7 Rational Exponents 261

 1

1

1
 3 1
65. Find the simplified form of 32 2  3 2  8 2 . 2 2  3 2
4 Assess 1

66. What is the simplified form of 81 3  24 3  3 3 ? 2  3 3
1

1

1


Open-Ended Assessment MUSIC For Exercises 67 and 68, use the following information.
On a piano, the frequency of the A note above middle C should be set at
Speaking Have students write
440 vibrations per second. The frequency fn of a note that is n notes above
two expressions with rational ex- n
that A should be fn  440  2 12 .
ponents, one that is in simplified
67. At what frequency should a piano tuner set the A that is one octave, or 12 notes,
form and another that is not. Ask
above the A above middle C? 880 vibrations per second
them to explain the difference
68. Middle C is nine notes below the A that has a frequency of 440 vibrations per
between them, using the four second. What is the frequency of middle C? about 262 vibrations per second
conditions listed in the Concept
Summary on p. 260. 69. BIOLOGY Suppose a culture has 100 bacteria to begin with and the number of
bacteria doubles every 2 hours. Then the number N of bacteria after t hours is
t

given by N  100  2 2 . How many bacteria will be present after 3 and a half
hours? about 336
Intervention
New In order to help 70. CRITICAL THINKING Explain how to solve 9x  3x  2 for x. See margin.
1

students see
why the excep- Music 71. WRITING IN MATH Answer the question that was posed at the beginning of
tion “except The first piano was the lesson. See pp. 283A–283B.
made in about 1709 by
when b  0 and n is even” is Bartolomeo Cristofori, a How do rational exponents apply to astronomy?
necessary when defining maker of harpsichords Include the following in your answer:
Mp 2
rational exponents, ask them in Florence, Italy. • an explanation of how to write the formula r  D 5 in radical form and
Source: www.infoplease.com MS
to choose values for b and n simplify it, and
that violate these constraints, • an explanation of what happens to the value of r as the value of D increases
assuming that Mp and MS are constant.
and see what results when
applying the definition. 1 4
72. Which is the value of 4 2   ? C
1
Standardized 2
Test Practice A 1 B 2 C 2
1 D 2
1
16 2
Getting Ready for 73. If 4x  2y  5 and x  y  1, then what is the value of 3x  3y? C
Lesson 5-8 A 1 B 2 C 4 D 6
PREREQUISITE SKILL Lesson 5-8
presents solving equations and
inequalities that contain radicals. Maintain Your Skills
Solving such equations and
inequalities involves finding the Mixed Review Simplify. (Lessons 5-5 and 5-6)
power of an expression involving 4x3y2 2xyx
74.    362
75. 26 312
a radical. Exercises 81–84 should   18
76. 32  22
  50 77. 
4
(8)4 8
be used to determine your
students’ familiarity with 78. 4
(x  5
)2 4x  5 79. 396x4 12x2
multiplying radicals.
80. BIOLOGY Humans blink their eyes about once every 5 seconds. How many
times do humans blink their eyes in two hours? (Lesson 1-1) 1440
Assessment Options
Quiz (Lessons 5-6 and 5-7) is Getting Ready for PREREQUISITE SKILL Find each power. (To review multiplying radicals, see Lesson 5-6.)
available on p. 308 of the Chapter 5 the Next Lesson 81. 
x  2 x  2 82. 2x  3  2x  3
2 3 3
Resource Masters.
83. x  1 x  2x  1 84. 2x  3 4x  12x  9
2 2

262 Chapter 5 Polynomials

Answer
70. Rewrite the equation so that the bases are the same on each side.
1
x x  2
9 3
1
(32)x  3x  2
1
2x x  
3 3 2
Since the bases are the same and this is an equation, the exponents
1 1
must be equal. Solve 2x  x   . The result is x   .
2 2
262 Chapter 5 Polynomials
 Radical Equations Lesson
and Inequalities Notes

• Solve equations containing radicals.

• Solve inequalities containing radicals.
1 Focus
Vocabulary do radical equations apply to manufacturing?
• radical equation Computer chips are made from the element silicon, which is found in sand. 5-Minute Check
• extraneous solution Suppose a company that manufactures computer chips uses the formula Transparency 5-8 Use as a
• radical inequality 2
C  10n 3  1500 to estimate the cost C in dollars of producing n chips. quiz or review of Lesson 5-7.
This equation can be rewritten as a radical equation.
Mathematical Background notes
SOLVE RADICAL EQUATIONS Equations with radicals that have variables are available for this lesson on
in the radicands are called radical equations. To solve this type of equation, raise p. 220D.
each side of the equation to a power equal to the index of the radical to eliminate
the radical. do radical equations
apply to manufacturing?
Example 1 Solve a Radical Equation
Ask students:
Solve 
x  1  2  4. • Why can the equation be
x124 Original equation rewritten as a radical equation?

x12 Subtract 2 from each side to isolate the radical. because the variable n has a
x
 1   22
2
Square each side to eliminate the radical. rational exponent
x14 Find the squares.
• Manufacturing There are pro-
x3 Subtract 1 from each side.
duction costs associated with
CHECK 
x124 Original equation manufactured goods that occur

3124 Replace x with 3.
even before the first item is
44⻫ Simplify. made. That is, there is still a
cost even if no items have been
The solution checks. The solution is 3.
produced yet. How much are
these costs for the production
When you solve a radical equation, it is very important that you check your
solution. Sometimes you will obtain a number that does not satisfy the original of this company’s computer
equation. Such a number is called an extraneous solution . You can use a graphing chips? $1500
calculator to predict the number of solutions of an equation or to determine whether
the solution you obtain is reasonable.

Example 2 Extraneous Solution

Solve 
x  15  3  x.
x  15  3  x Original equation

x  15   3  x 
2 2
Square each side.
x  15  9  6x  x Find the squares.
24  6x Isolate the radical.
4  x Divide each side by 6.
42  x 
2
Square each side again.
16  x Evaluate the squares.
(continued on the next page)
Lesson 5-8 Radical Equations and Inequalities 263

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 5 Resource Masters 5-Minute Check Transparency 5-8
• Study Guide and Intervention, pp. 281–282 Answer Key Transparencies
• Skills Practice, p. 283
• Practice, p. 284 Technology
• Reading to Learn Mathematics, p. 285 Alge2PASS: Tutorial Plus, Lesson 9
• Enrichment, p. 286 Interactive Chalkboard

Lesson x-x Lesson Title 263

 CHECK x  15  3  x
2 Teach 
16  15  3  16

 1  3  4
SOLVE RADICAL 1  1
EQUATIONS
The solution does not check, so the equation
In-Class Examples Power
Point®
has no real solution.
The graphing calculator screen shows the
1 Solve 
y  2  1  5. 38 graphs of y   x  15 and y  3  x. The
graphs do not intersect, which confirms that
2 Solve x
 12  2  x. there is no solution. [10, 30] scl: 5 by [5, 5] scl: 1

no solution
Teaching Tip Remind students
that the square root sign in an You can apply the same methods used in solving square root equations to solving
equation means the principal equations with roots of any index. Remember that to undo a square root, you square
root. the expression. To undo an nth root, you must raise the expression to the nth power.
1

3 Solve (3y  1) 3  5  0. 42 Study Tip Example 3 Cube Root Equation
Teaching Tip Have a Alternative Solve 3(5n  1) 3  2  0.
1

discussion with students about Method 1

which operations may introduce To solve a radical In order to remove the  power, or cube root, you must first isolate it and then
3
equation, you can raise each side of the equation to the third power.
extraneous solutions when substitute a variable for
solving a radical equation. the radical expression. In 1
Example 3, let A  5n  1.
3(5n  1) 3  2  0 Original equation
1
1
3A  2  0
3 3(5n  1) 3  2 Add 2 to each side.
1 2 1
3A  2
3
(5n  1)   3 Divide each side by 3.
1
2 3
A 3  
3 1 3 2 3
8
A  
27
(5n  1) 3  
3   Cube each side.

8
5n  1   8
27 5n  1   Evaluate the cubes.
27
7
n   35
27
5n   Add 1 to each side.
27
7
n 
 Divide each side by 5.
27

1
CHECK 3(5n  1) 3  2  0 Original equation
1
 7
3 5    1 3  2  0
27  7
Replace n with .
27
8 1
3  3  2  0
27   Simplify.

2
3   2  0
3   8 2
The cube root of  is .
27 3
0  0 ⻫ Subtract.
7
The solution is .
27

SOLVE RADICAL INEQUALITIES You can use what you know about radical
equations to help solve radical inequalities. A radical inequality is an inequality
that has a variable in a radicand.
264 Chapter 5 Polynomials

264 Chapter 5 Polynomials

 Example 4 Radical Inequality SOLVE RADICAL
Solve 2  
4x  4  6.
INEQUALITIES
Since the radicand of a square root must be greater than or equal to zero, first In-Class Example Power
Point®
solve 4x  4  0 to identify the values of x for which the left side of the given
inequality is defined.
4 Solve 
3x  6  4  7.
4x  4  0
2x5
4x  4
x1
Teaching Tip Emphasize the
Now solve 2  
4x  4  6. importance of checking key test
2  
4x  4  6 Original inequality
values in the appropriate ranges.


4x  4  4 Isolate the radical.
4x  4  16 Eliminate the radical.
4x  20
x5
Add 4 to each side.
Divide each side by 4.
3 Practice/Apply
It appears that 1  x  5. You can test some x values to confirm the solution.
Let f(x)  2   4x  4. Use three test values: one less than 1, one between
1 and 5, and one greater than 5. Organize the test values in a table. Study Notebook
x0 x2 x7 Have students—
f(0)  2  
4(0) 
4 
f(2)  2  4(2) 
4 
f(7)  2  4(7) 
4 • add the definitions/examples of

 2  4 4  6.90 the vocabulary terms to their
Since 4  is not a Since 4  6, the Since 6.90  6, the
Vocabulary Builder worksheets for
real number, the inequality is satisfied. inequality is not
Study Tip inequality is not satisfied. satisfied. Chapter 5.
Check Your • write a list of the steps for solving
Solution The solution checks. Only values in the interval 1  x  5 satisfy the inequality.
You may also want to use You can summarize the solution with a number line. radical equations and copy the list
a graphing calculator to of steps for solving radical
check. Graph each side of
2 1 0 1 2 3 4 5 6 7 8
the original inequality and inequalities given in the Concept
examine the intersection.
Summary on p. 265.
• include any other item(s) that they
Solving Radical Inequalities find helpful in mastering the skills
To solve radical inequalities, complete the following steps.
in this lesson.
Step 1 If the index of the root is even, identify the values of the variable for which
the radicand is nonnegative.
Step 2 Solve the inequality algebraically.
Step 3 Test values to check your solution.
Answers
1. Since x is not under the radical,
the equation is a linear equation,
not a radical equation. The
Concept Check 1. Explain why you do not have to square each side to solve 2x  1  3. 3  1
solution is  .
Then solve the equation. See margin. 2
2. Show how to solve x  6x  9  0 by factoring. Name the properties of 2. The trinomial is a perfect square
equality that you use. See margin. in terms of x. x  6x  9 
3. OPEN ENDED Write an equation containing two radicals for which 1 is a (x  3)2, so the equation can
solution. Sample answer: x  
x33 2
be written as (x  3)  0.
www.algebra2.com/extra_examples Lesson 5-8 Radical Equations and Inequalities 265
Take the square root of each side
to get x  3  0. Use the
Addition Property of Equality to
Differentiated Instruction add 3 to each side, then square
each side to get x  9.
Logical Have students compare solving radical equations and
inequalities to solving other types of equations and inequalities. Have
them write or give a short presentation about the similarities and
differences between the procedures used in the solution processes.

Lesson 5-8 Radical Equations and Inequalities 265

 Study
5-8 Guide
NAME ______________________________________________ DATE

andIntervention
Intervention,
____________ PERIOD _____
Guided Practice Solve each equation or inequality.
Study Guide and 1
p. 281
Radical(shown)
Equations andand p. 282
Inequalities GUIDED PRACTICE KEY 4. 
4x  1  3 2 5. 4  (7  y) 2  0 9
Solve Radical Equations The following steps are used in solving equations that have
variables in the radicand. Some algebraic procedures may be needed before you use these
steps.
Exercises Examples 6. 1  
x  2  0 no solution 7. 
z  6  3  0 15
4–9, 12 1–3 1 1
9. 
Step 1 Isolate the radical on one side of the equation.
3
Step
Step
2
3
To eliminate the radical, raise each side of the equation to a power equal to the index of the radical.
Solve the resulting equation.
10, 11 4 8. (12a) 3  1 18 x  4  3 31
Step 4 Check your solution in the original equation to make sure that you have not obtained any extraneous roots. 6
3
10. 
2x  3  4  5   x  39 11. 
b  12  b 2 0  b 4
Example 1 Solve 2   4  8.
4x  8 Example 2 Solve    5x
3x  1   1. 2
2
4x  8  4  8 Original equation 3x  1  5x 1 Original equation
24x  8  12 Add 4 to each side. 3x  1  5x  2 5x  1 Square each side.
 25x
  2x
4x  8  6
4x  8  36
4x  28
Isolate the radical.
Square each side. 5x
x
5x  x2
Simplify.
Isolate the radical. Application 12. GEOMETRY The surface area S of a cone can be found
by using S  r r2  h2, where r is the radius of the
Subtract 8 from each side. Square each side.

S  225 cm2

Lesson 5-8
x7 Divide each side by 4. x2  5x  0 Subtract 5x from each side.
Check x(x  5)  0 Factor.

2
4(7) 
848 x  0 or x  5
Check
base and h is the height of the cone. Find the height of h
236
48
2(6)  4  8 3(0)  1  1, but 5(0)
not a solution.
  1  1, so 0 is the cone. about 13.42 cm
88
3(5)  1  4, and 5(5)
  1  4, so the
The solution x  7 checks. solution is x  5.
r  5 cm
Exercises
Solve each equation.

1. 3  2x3
5 2. 2
3x  4  1  15 3. 8  
x12 ★ indicates increased difficulty
3

 15 no solution

4. 
3

5x46 5. 12  
2x  1  4 6.  0
12  x
Practice and Apply
95 no solution 12
Homework Help Solve each equation or inequality.
7. 21
  
5x  4  0 8. 10  2x
5 9.    
x2  7x 7x  9
For See
5 12.5 no solution
Exercises Examples 13. x  4 16 14. y  7  0 49
10. 4
3
  2  10
2x  11 11. 2
x  11  
x  2  
3x  6 12.  x1
9x  11 13–24, 1–3 1 1
8 14 3, 4 29–32, 15. a  9  0 no solution
2 16. 2  4z 2  0 no solution
NAME ______________________________________________ DATE ____________
GlPERIOD
Al _____
37–42
17.  18.  3 5
Gl /M G Hill 281 b 2 3 3
Skills
5-8 Practice,
Practice (Average)
p. 283 and 25–28, 4 c12 9 5m  2
Practice, p. 284 33–36
Radical Equations and(shown)
Inequalities 27
Solve each equation or inequality. 19. 7  
4x  8  9 1 20. 5   4y  5  12 
Extra Practice 2
1. x
  8 64 2. 4  x
3 1
1 1

49 1
See page 839. 21. (6n  5)  3  2 20 3 22. (5x  7) 5  3  5 5
3. 2p
  3  10  4. 43h
20 
2 12

23. 
x  5  
2x  4 no solution 24. 
2t  7  
t2 9
1 1
 
5. c  6  9 9
2 6. 18  7h  12 no solution
2

3 5
7.  8. 
25. 1   26. 
d  2  7 341 w71 8

3 4
7x  3 3 x 1 3x  6  2  5 2  x  1
9. 6  
q  4  9 31 10. 
y  9  4  0 no solution

11. 
2m 
6  16  0 131 12. 
3
4m 
1 22 
63
4
27. 2  
9  5x  6 x  11 28. 6  
2y  1  3 y 4

★ 29. 
x  6  x  3 no solution ★ 30. 
y  21  1  
7 3
1  4t  8  6  
y  12 4
13.  12 
8n  5 14. 
4 4
1
41 
15. 
2t  5  3  3  16. (7v  2) 4  12  7 no solution
2
1

17. (3g  1) 2  6  4 33
1

18. (6u  5) 3  2  3 20
★ 31. 
b  1  
b61 3 ★ 32. 
4z  1  3  
4z  2 no solution
19.    
2d  5 d1 4 20. 
4r  6  r 2
★ 33. 2  
x  6  x 0  x  2 ★ 34. 
a  9  a 3
 0a 3
7
21. 
6x  4    
2x  10 22. 
2x  5  
2x  1 no solution
79
★ 35. 
b  5  
b74 b5 ★ 36. 
c  5  
c  10 2.5 c 
2

  12 a  16
23. 3a z  5  4  13 5  z  76
24.  16
37. What is the solution of 2  
3
25. 8  2q
  5 no solution 26.    5  a 14
2a  3
2 x  6  1? 3
c46 c5 x  1  2 x 7
3
27. 9   28. 
38. Solve 
2x  4  4  2. 16
29. STATISTICS Statisticians use the formula   v  to calculate a standard deviation ,
where v is the variance of a data set. Find the variance when the standard deviation
is 15. 225

30. GRAVITATION Helena drops a ball from 25 feet above a lake. The formula 39. CONSTRUCTION The minimum depth d in inches of a beam required
1
t     describes the time t in seconds that the ball is h feet above the water.
25  h

s
4
How many feet above the water will the ball be after 1 second? 9 ft
to support a load of s pounds is given by the formula d  ,
576w
where  is the length of the beam in feet and w is the width in feet. Find
Gl NAME
/M G ______________________________________________
Hill 284 DATE ____________
Gl PERIOD
Al _____
b 2
Reading
5-8 Readingto
to Learn
Learn Mathematics
Mathematics, p. 285 ELL the load that can be supported by a board that is 25 feet long, 2 feet wide,
Radical Equations and Inequalities
Pre-Activity How do radical equations apply to manufacturing? and 5 inches deep. 1152 lb
Read the introduction to Lesson 5-8 at the top of page 263 in your textbook.
Explain how you would use the formula in your textbook to find the cost of
producing 125,000 computer chips. (Describe the steps of the calculation in the
order in which you would perform them, but do not actually do the calculation.)
2
40. AEROSPACE ENGINEERING The radius r of the orbit of a satellite is given


Sample answer: Raise 125,000 to the  power by taking the 2
3 GMt
3
cube root of 125,000 and squaring the result (or raise 125,000 by r  , where G is the universal gravitational constant, M is the mass
2
to the  power by entering 125,000 ^ (2/3) on a calculator).
3
4 2
Multiply the number you get by 10 and then add 1500. of the central object, and t is the time it takes the satellite to complete one orbit.
Solve this formula for t.
42r 3

Reading the Lesson
1. a. What is an extraneous solution of a radical equation? Sample answer: a number t  
that satisfies an equation obtained by raising both sides of the original
equation to a higher power but does not satisfy the original equation
266 Chapter 5 Polynomials GM
b. Describe two ways you can check the proposed solutions of a radical equation in order
to determine whether any of them are extraneous solutions. Sample answer: One
way is to check each proposed solution by substituting it into the
original equation. Another way is to use a graphing calculator to graph NAME ______________________________________________ DATE ____________ PERIOD _____
both sides of the original equation. See where the graphs intersect.
This can help you identify solutions that may be extraneous.
2. Complete the steps that should be followed in order to solve a radical inequality.
Enrichment,
5-8 Enrichment p. 286
Step 1 If the index of the root is even , identify the values of
Truth Tables
the variable for which the radicand is nonnegative .
In mathematics, the basic operations are addition, subtraction, multiplication,
Step 2 Solve the inequality algebraically. division, finding a root, and raising to a power. In logic, the basic operations
are the following: not (), and (), or (), and implies (→).
Step 3 Test values to check your solution.
If P and Q are statements, then P means not P; Q means not Q; P  Q
means P and Q; P  Q means P or Q; and P → Q means P implies Q. The
Helping You Remember operations are defined by truth tables. On the left below is the truth table for
the statement P. Notice that there are two possible conditions for P, true (T)
3. One way to remember something is to explain it to another person. Suppose that your
or false (F). If P is true, P is false; if P is false, P is true. Also shown are the
friend Leora thinks that she does not need to check her solutions to radical equations by
truth tables for P  Q, P  Q, and P → Q.
substitution because she knows she is very careful and seldom makes mistakes in her
work. How can you explain to her that she should nevertheless check every proposed P P P Q PQ P Q PQ P Q P→Q
solution in the original equation? Sample answer: Squaring both sides of an
equation can produce an equation that is not equivalent to the original T F T T T T T T T T T
one. For example, the only solution of x  5 is 5, but the squared F T T F F T F T T F F
equation x2  25 has two solutions, 5 and 5. F T F F T T F T T
F F F F F F F F T

You can use this information to find out under what conditions a complex
statement is true.

266 Chapter 5 Polynomials

 41. PHYSICS When an object is dropped from the top of a 50-foot tall
building, the object will be h feet above the ground after t seconds,

50 h
About the Exercises…
where   t. How far above the ground will the object be after
4
1 second? 34 ft
Organization by Objective
• Solve Radical Equations:
42. HEALTH Use the information about health at the left. 13–24, 29–32, 37–42
A 70-kilogram person who is 1.8 meters tall has a ponderal index of about • Solve Radical Inequalities:
2.29. How much weight could such a person gain and still have an index of 25–28, 33–36
at most 2.5? 21.125 kg
Odd/Even Assignments
43. CRITICAL THINKING Explain how you know that  x  2  
2x  3  1 Exercises 13–38 are structured
has no real solution without trying to solve it. See margin. so that students practice the
Health same concepts whether they
A ponderal index p is a 44. WRITING IN MATH Answer the question that was posed at the beginning
measure of a person’s
are assigned odd or even
of the lesson. See pp. 283A–283B.
body based on height h problems.
in meters and mass m How do radical equations apply to manufacturing?
in kilograms. One such Include the following in your answer:
formula is p  
.
m
3
2
• the equation C  10n 3  1500 rewritten as a radical equation, and
Assignment Guide
h
Source: A Dictionary of Food • a step-by-step explanation of how to determine the maximum number Basic: 13–27 odd, 37–41 odd,
and Nutrition of chips the company could make for $10,000. 43–59
Average: 13–41 odd, 43–59
Standardized 45. If 
x  5  1  4, what is the value of x? D Advanced: 14–42 even, 43–53
Test Practice A 4 B 0 C 2 D 4 (optional: 54–59)
46. Side 
AC of triangle ABC contains which of the y B
following points? C
A

D
(3, 4)

(4, 5)
B

E
(3, 5)

(4, 6)
C (4, 3) 8

10
C 4 Assess
A (0, 0)
O x Open-Ended Assessment
Writing Have students write a
list of examples showing how to
Maintain Your Skills solve each of the different types of
radical equations and inequalities
Mixed Review Write each radical using rational exponents. (Lesson 5-7)
discussed in this lesson.
3 1  2
47.  48.  49. x2  1  (x2  1) 3
2
x  7 (x  7) 2
7 3
53 5 7

Simplify. (Lesson 5-6)

3
72x6y3 6x3y
50.  2y 51.  1


100 
52. 5  3  28  103
2 Intervention
10
3
10 New Make sure that
53. BUSINESS A dry cleaner ordered 7 drums of two different types of cleaning students know
fluid. One type cost $30 per drum, and the other type cost $20 per drum. The the constraints
total cost was $160. How much of each type of fluid did the company order? on the values
Write a system of equations and solve by graphing. (Lesson 3-1)
x  y  7, 30x  20y  160; See margin for graph; (2, 5). of the variables in a radical
equation so that the solutions
Getting Ready for PREREQUISITE SKILL Simplify each expression. are real numbers.
the Next Lesson (To review binomials, see Lesson 5-2.)
54. (5  2x)  (1  x) 4  x 55. (3  2y)  (4  y) 1  y
56. (4  x)  (2  3x) 2  4x 57. (7  3x)  (4  3x) 11
58. (1  z)(4  2z) 4  6z  2z2 59. (3  4x)(1  2x) 3  10x  8x2
Getting Ready for
www.algebra2.com/self_check_quiz Lesson 5-8 Radical Equations and Inequalities 267
Lesson 5-9
PREREQUISITE SKILL Lesson 5-9
presents calculating with complex
Answers 53. y numbers, often written in the form
30x  20y  160 of binomials. Exercises 54–59
43. Since  x  2  0 and  2x  3  0, should be used to determine
the left side of the equation is (2, 5) your students’ familiarity with
nonnegative. Therefore, the left side x y 7 adding, subtracting, and
of the equation cannot equal 1. multiplying binomials.
Thus, the equation has no solution.

O x

Lesson 5-8 Radical Equations and Inequalities 267

Graphing
Calculator A Follow-Up of Lesson 5-8
Investigation
A Follow-Up of Lesson 5-8

Getting Started Solving Radical Equations

and Inequalities by Graphing
Know Your Calculator The TI-83 You can use a TI-83 Plus to solve radical equations and inequalities. One way to do
Plus automatically supplies a left this is by rewriting the equation or inequality so that one side is 0 and then using
parenthesis after each radical sign. the zero feature on the calculator.
When functions are entered on
the Y= list, it is important to sup-   
Solve x x  2  3.
ply right parentheses as needed
to ensure correct graphs and Rewrite the equation. Use a table.
correct numerical results. • Subtract 3 from each side of the equation to • You can use the TABLE function to locate
obtain x  
x  2  3  0. intervals where the solution(s) lie. First, enter
Displaying Tables In Step 2 on the starting value and the interval for the table.
p. 268, students should check to • Enter the function y  x  
x  2  3 in
KEYSTROKES: 2nd [TBLSET] 0 ENTER 1 ENTER
the Y= list.
be sure that the AUTO option has
KEYSTROKES: Review entering a function on
been selected on each of the last
page 128.
two lines of the TABLE SETUP screen.
Exact Solutions The approximate
zero displayed on the screen
shown in Step 4 on p. 268 appears
to be a repeating decimal. If you
go to the home screen immedi-
ately after Step 4 and use the key- Estimate the solution. Use the zero feature.
strokes X,T,,n 1 ENTER , • Complete the table and estimate the • Graph, then select zero from the CALC menu.
the calculator will display a solution(s). KEYSTROKES: GRAPH 2nd [CALC] 2
49 KEYSTROKES: 2nd [TABLE]
fraction for the exact solution,  .
36
The calculator can be used to
verify that this is indeed the
exact solution of the equation.

Teach [10, 10] scl: 1 by [10, 10] scl: 1

Since the function changes sign from negative Place the cursor to the left of the zero and press
• After reading the sentence at to positive between x  1 and x  2, there is a ENTER for the Left Bound. Then place the
solution between 1 and 2. cursor to the right of the zero and press ENTER
the top of p. 269, have students
solve the radical equation on for the Right Bound. Press ENTER to solve.
p. 268 again treating each side as The solution is about 1.36. This agrees with
a separate function. Point out the estimate made by using the TABLE.
that the right side will simply be
graphed as the function y  3. www.algebra2.com/other_calculator_keystrokes
• After completing the discussion
of the procedure on p. 269 for
268 Chapter 5 Polynomials
solving a radical inequality, have
students solve it again by first
subtracting 2x from both sides
and then graphing the function
y   x  2  1  2x. Point
out that the portion of the
graph below the x-axis shows
the solution.
• Have students complete
Exercises 1–10.

268 Chapter 5 Polynomials

 Instead of rewriting an equation or inequality so that one side is 0, you can also treat
each side of the equation or inequality as a separate function and graph both.
Assess
 
Solve 2x x  2  1.
• When examining the table of
Graph each side of the inequality. Use the trace feature. values for the equation on
• In the Y= list, enter y1  2x and • Press TRACE . You can use or to p. 268, why do you know a
y2  
x  2  1. Then press GRAPH . switch the cursor between the two curves. solution lies between the two
x values where the graphed
function changes signs?
Sample answer: Since the original
equation was solved so that one side
was zero, the function representing
the other side has a value of 0 when
the value of y is 0. That must occur
[10, 10] scl: 1 by [10, 10] scl: 1 [10, 10] scl: 1 by [10, 10] scl: 1 for a value of x somewhere between
two values of x whose correspond-
The calculator screen above shows that, for
ing values of y have different signs
points to the left of where the curves cross,
(because 0 lies between the
Y1  Y2 or 2x   x  2  1. To solve the
original inequality, you must find points for positive numbers and the negative
which Y1 Y2. These are the points to the numbers).
right of where the curves cross.

Use the intersect feature. Answer

• You can use the INTERSECT feature on the CALC menu to approximate 10. Rewrite the inequality so that one
the x-coordinate of the point at which the curves cross.
side is 0. Then graph the other side
KEYSTROKES: 2nd [CALC] 5
and find the x values for which the
• Press ENTER for each of First curve?, Second curve?, and Guess?. graph is above or below the x-axis,
The calculator screen shows that the x-coordinate of the point at according to the inequality symbol.
which the curves cross is about 2.40. Therefore, the solution of the Use the zero feature to approxi-
inequality is about x 2.40. Use the symbol instead of  in the [10, 10] scl: 1 by [10, 10] scl: 1 mate the x-coordinate of the point
solution because the symbol in the original inequality is .
at which the graph crosses the
x-axis.
Exercises 4. about 3.89 5. about 2.52 8. about 0  x 1 9. about 1  x 4.52
Solve each equation or inequality.
1.  x43 5 2. 3x  5  1 2 3. x  5  3x  4 0.5
4. x 
 3  x 24 5. 3
x  7  2
x21 6. x  8 1  x  2 4.25
7. 
x32 x7 8. 
x  3 2x 9. x  x 14
10. Explain how you could apply the technique in the first example to solving
an inequality. See margin.

Graphing Calculator Investigation Solving Radical Equations and Inequalities by Graphing 269

Graphing Calculator Investigation Solving Radical Equations and Inequalities by Graphing 269
 Lesson Complex Numbers
Notes

• Add and subtract complex numbers.

1 Focus • Multiply and divide complex numbers.

Vocabulary do complex numbers apply to polynomial equations?

5-Minute Check • imaginary unit Consider the equation 2x2  2  0. If you solve this equation for x2, the result is
Transparency 5-9 Use as a • pure imaginary number x2  1. Since there is no real number whose square is 1, the equation has no
quiz or a review of Lesson 5-8. • complex number real solutions. French mathematician René Descartes (1596–1650) proposed that
• absolute value a number i be defined such that i2  1.
Mathematical Background notes • complex conjugates
are available for this lesson on ADD AND SUBTRACT COMPLEX NUMBERS Since i is defined to have
p. 220D. the property that i2  1, the number i is the principal square root of 1; that is,
i  1. i is called the imaginary unit. Numbers of the form 3i, 5i, and i2 are
do complex numbers called pure imaginary numbers. Pure imaginary numbers are square roots of
apply to polynomial negative real numbers. For any positive real number b,  b2   b2 or bi.
equations?
Example 1 Square Roots of Negative Numbers
Ask students:
Simplify.
• If by definition i2  1, then Study Tip

a. 18 5
b. 125x
what do you think is the value Reading Math 
18  
1  32
2 
125x5  
1  52
 x4 
5x
of i4? Justify your answer. i is usually written before
radical symbols to make it   
 1 32  2  
1  
52  
x4  
5x
Since i 4  (i 2)2, replacing i 2 with clear that it is not under  i  3  2 or 3i2  or 5ix25x
 i  5  x2  5x 
1 gives i 4  (1)2 or 1. the radical.

The Commutative and Associative Properties of Multiplication hold true for pure
TEACHING TIP
imaginary numbers.
2 Teach Point out that when
multiplying radicals
with negative radicands,
Example 2 Multiply Pure Imaginary Numbers
students should first take Simplify.
ADD AND SUBTRACT the roots, then multiply.
a. 2i  7i   15
b. 10 
COMPLEX NUMBERS Otherwise, their answers
may be off by a factor 2i  7i  14i2    i10
10 15   i15

In-Class Examples Power of 1.  14(1) i 2  1 
 i2150
Point®
 14   6
 1  25
1 Simplify. 
 56

a. 28
 2i 7 You can use the properties of powers to help simplify powers of i.
b. 
32y3 4i | y | 2y
 Example 3 Simplify a Power of i
2 Simplify. Simplify i45.
a. 3i  2i 6 i45  i  i44 Multiplying powers
i (i2)22 Power of a Power
b. 12
  2
 26  i  (1)22 i 2  1
 i  1 or i (1)22  1
3 Simplify i35. i
270 Chapter 5 Polynomials

Resource Manager
Workbook and Reproducible Masters Transparencies
Chapter 5 Resource Masters School-to-Career Masters, p. 10 5-Minute Check Transparency 5-9
• Study Guide and Intervention, pp. 287–288 Teaching Algebra With Manipulatives Answer Key Transparencies
• Skills Practice, p. 289 Masters, pp. 239, 240
• Practice, p. 290 Technology
• Reading to Learn Mathematics, p. 291 Interactive Chalkboard
• Enrichment, p. 292
• Assessment, p. 308
 The solutions of some equations involve pure imaginary numbers.
In-Class Examples Power
Point®
Example 4 Equation with Imaginary Solutions
4 Solve 5y2  20  0. 2i
Solve 3x2  48  0.
3x2  48  0 Original equation Teaching Tip Make sure
3x2  48 students understand that when
Study Tip Subtract 48 from each side.
they take the square root of
Quadratic x2  16 Divide each side by 3.
both sides of an equation, they
Solutions  Take the square root of each side.
x  16 must use the  symbol in front
Quadratic equations
x  4i   16
16   1
 of the radical sign.
always have complex
solutions. If the
discriminant is: 5 Find the values of x and y
• negative, there are two that make the equation
imaginary roots,
• zero, there are two Consider an expression such as 5  2i. Since 5 is a real number and 2i is a pure 2x  yi  14  3i true.
equal real roots, or imaginary number, the terms are not like terms and cannot be combined. This type x  7, y  3
• positive, there are two of expression is called a complex number .
unequal real roots. Teaching Tip Emphasize that
two complex numbers are equal
Complex Numbers if and only if their real parts are
equal and their imaginary parts
• Words A complex number is any number that can be written in the form
a  bi, where a and b are real numbers and i is the imaginary unit. are equal.
a is called the real part, and b is called the imaginary part.
• Examples 7  4i and 2  6i  2  (6)i

The Venn diagram at the right

shows the complex number system. Complex Numbers (a  bi )

• If b  0, the complex number is a Real Numbers Imaginary Numbers

real number. b0 (b  0)

• If b  0, the complex number is Pure

imaginary. Imaginary
Numbers
• If a  0, the complex number is a a0
pure imaginary number.

Two complex numbers are equal if and only if their real parts are equal and their
imaginary parts are equal. That is, a  bi  c  di if and only if a  c and b  d.

Example 5 Equate Complex Numbers

Find the values of x and y that make the equation
2x  3  (y  4)i  3  2i true.
Set the real parts equal to each other and the imaginary parts equal to each other.
Study Tip 2x  3  3 Real parts
2x  6 Add 3 to each side.
Reading Math
The form a  bi is x  3 Divide each side by 2.
sometimes called the
standard form of a y42 Imaginary parts
complex number.
y6 Add 4 to each side.

www.algebra2.com/extra_examples Lesson 5-9 Complex Numbers 271

Differentiated Instruction ELL

Verbal/Linguistic Have students write poems about the imaginary
number i and the repeating values of its powers, perhaps including
wordplay with the terms real and imaginary. The content of the poems
should be helpful for remembering the mathematical characteristics of i.

Lesson 5-9 Complex Numbers 271

 To add or subtract complex numbers, combine like terms. That is, combine the
In-Class Example Power
Point® real parts and combine the imaginary parts.

6 Simplify.
Example 6 Add and Subtract Complex Numbers
a. (3  5i)  (2  4i) 5  i Simplify.
b. (4  6i)  (3  7i) 1  i a. (6  4i)  (1  3i)
(6  4i)  (1  3i)  (6  1)  (4  3)i Commutative and Associative Properties
7i Simplify.
Answers
b. (3  2i)  (5  4i)
Algebra Activity (3  2i)  (5  4i)  (3  5)  [2  (4)]i Commutative and Associative Properties
1. imaginary b  2  2i Simplify.

4
1
(3  2i ) You can model the addition and subtraction of complex numbers geometrically.
(1  i )
O real a

Adding Complex Numbers

You can model the addition of complex numbers on a coordinate plane.
The horizontal axis represents the real part a of the complex number, and
2. Rewrite the difference as a sum, the vertical axis represents the imaginary part b of the complex number.
(3  2i)  (4  i)  (3  2i)
 (4  i). Then apply the Use a coordinate plane to find (4  2i )  (2  3i ).
method discussed in this activity. • Create a coordinate plane and label the axes b imaginary
appropriately. 2  5i
• Graph 4  2i by drawing a segment from the
origin to (4, 2) on the coordinate plane. 3
2
• Represent the addition of 2  3i by moving 4  2i
2 units to the left and 3 units up from (4, 2).
O real a
• You end at the point (2, 5), which represents
the complex number 2  5i.
So, (4  2i )  (2  3i )  2  5i.

Model and Analyze

1. Model (3  2i )  (4  i ) on a coordinate plane. See margin.
2. Describe how you could model the difference (3  2i )  (4  i ) on a
coordinate plane. See margin.
3. The absolute value of a complex number is the distance from the origin to
the point representing that complex number in a coordinate plane. Refer to
the graph above. Find the absolute value of 2  5i. 29

4. Find an expression for the absolute value of a  bi. 
a 2  b 2

MULTIPLY AND DIVIDE COMPLEX NUMBERS Complex numbers

are used with electricity. In a circuit with alternating current, the voltage,
current, and impedance, or hindrance to current, can be represented by complex
numbers.
272 Chapter 5 Polynomials

Algebra Activity
Materials: grid paper, ruler/straightedge
• The horizontal axis is often called a real number line. What might be a
corresponding name for the vertical axis? an imaginary number line
• Where do real numbers lie on this coordinate plane? on the horizontal axis
Where do pure imaginary numbers lie? on the vertical axis
• Where do complex numbers for which neither a nor b is 0 lie on this
coordinate plane? on the regions of the plane other than the axes
272 Chapter 5 Polynomials
 You can use the FOIL method to multiply complex numbers.
MULTIPLY AND DIVIDE
COMPLEX NUMBERS
Example 7 Multiply Complex Numbers
ELECTRICITY In an AC circuit, the voltage E, current I, and impedance Z are In-Class Examples Power
Point®
related by the formula E  I  Z. Find the voltage in a circuit with current
Study Tip 1  3j amps and impedance 7  5j ohms. 7 ELECTRICITY In an AC circuit,
Reading Math EIZ Electricity formula
Electrical engineers use j the voltage E, current I, and
as the imaginary unit to  (1  3j)  (7  5j) I  1  3j, Z  7  5j impedence Z are related by
avoid confusion with the I  1(7)  1(5j)  (3j)7  3j(5j) FOIL the formula E  I  Z. Find the
for current.
 7  5j  21j  15j2 Multiply. voltage in a circuit with cur-
 7  16j  15(1) j 2  1 rent 1  4j amps and impe-
 22  16j Add. dence 3  6j ohms. 27  6j
The voltage is 22  16j volts.
8 Simplify.
5i 10 15
Two complex numbers of the form a  bi and a  bi are called complex a.    i
3  2i 13 13
conjugates . The product of complex conjugates is always a real number. For
example, (2  3i)(2  3i)  4  6i  6i  9 or 13. You can use this fact to simplify 4i 1 4
b.   
5
 i
5
the quotient of two complex numbers. 5i

Example 8 Divide Complex Numbers

Simplify.
3i
a. 
2  4i
5i
b. 
2i
3 Practice/Apply
i
3i 3i 2  4i 5i 5i i Why multiply by 
     2  4i and 2  4i      i
2  4i 2  4i 2  4i 2i 2i i 2i
are conjugates. intead of ?
2i
6i  12i2
  Multiply.
5i  i2
  Multiply.
Study Notebook
416i
2 2 2i
6i  12 5i  1
Have students—
  i 2  1   i 2  1
20 2 • complete the definitions/examples
3 3
   i Standard form
1 5
     i Standard form for the remaining terms on their
5 10 2 2
Vocabulary Builder worksheets for
Chapter 5.
• write a list of the first four powers
of i: i 1  1, i 2  1, i 3 
Concept Check 1. Determine if each statement is true or false. If false, find a counterexample.
i, i  1.
4

a. Every real number is a complex number. true • include any other item(s) that they
Study Tip find helpful in mastering the skills
b. Every imaginary number is a complex number. true
Look Back in this lesson.
Refer to Chapter 1 to
2. Decide which of the properties of a field and the properties of equality that the
review the properties of set of complex numbers satisfies. all of them
fields and the properties 3. OPEN ENDED Write two complex numbers whose product is 10.
of equality. Sample answer: 1  3i and 1  3i
Guided Practice Simplify.
GUIDED PRACTICE KEY  6i
4. 36 5. 
50x2 
y2 5ixy2
Exercises Examples 6. (6i)(2i) 12  1803
  318
7. 524
4, 5 1 8. i29 i 9. (8  6i)  (2  3i) 6  3i
6, 7 2
3i 7 11
8 3 10. (3  5i)(4  6i) 42  2i 11.    i
1  4i 17 17
9 6
10, 11 7, 8 Lesson 5-9 Complex Numbers 273

Lesson 5-9 Complex Numbers 273

 Study
NAME ______________________________________________ DATE

Guide andIntervention
Intervention,
____________ PERIOD _____
Solve each equation.
5-9 Study Guide and
p. 287 (shown)
Complex Numbers and p. 288 GUIDED PRACTICE KEY 12. 2x2  18  0 3i  14. 5x2  25  0 i 5
13. 4x2  32  0 2i 2
Add and Subtract Complex Numbers
Exercises Examples
Complex Number
A complex number is any number that can be written in the form a  bi,
where a and b are real numbers and i is the imaginary unit (i 2  1).
Find the values of m and n that make each equation true.
12–14 4
15. 2m  (3n  1)i  6  8i 3, 3
a is called the real part, and b is called the imaginary part.
Addition and
Subtraction of
Combine like terms.
(a  bi)  (c  di)  (a  c)  (b  d )i 15, 16 5 16. (2n  5)  (m  2)i  3  7i 5, 4
Complex Numbers (a  bi)  (c  di)  (a  c)  (b  d )i

Example 1
(6  i)  (4  5i)
Simplify (6  i)  (4  5i). Example 2
(8  3i)  (6  2i)
Simplify (8  3i)  (6  2i). Application 17. ELECTRICITY The current in one part of a series circuit is 4  j amps. The
 (6  4)  (1  5)i
 10  4i
 (8  6)  [3  (2)]i
 2  5i
current in another part of the circuit is 6  4j amps. Add these complex
To solve a quadratic equation that does not have real solutions, you can use the fact that numbers to find the total current in the circuit. 10  3j amps
i2  1 to find complex solutions.

Example 3 Solve 2x2  24  0.

★ indicates increased difficulty

2x2  24  0 Original equation
2x2  24 Subtract 24 from each side.
x2  12 Divide each side by 2.

x
x


12
2i
3
Take the square root of each side.
12
  4
  1
  3
 Practice and Apply

Lesson 5-9
Exercises
Simplify. Homework Help 
Simplify. 22. 132 30. 9  2i 33. 4  5i 35. 6  7i
For See
1. (4  2i)  (6  3i)
2i
2. (5  i)  (3  2i)
2i
3. (6  3i)  (4  2i)
10  5i Exercises Examples  12i
18. 144  9i
19. 81 20. 64x4 8x 2i
4. (11  4i)  (1  5i)
12  9i
5. (8  4i)  (8  4i)
16
6. (5  2i)  (6  3i)
11  5i
18–21 1 21. 
100a4
b2 10a2|b|i   26
22. 13  23. 6  24
 12
22–25 2
7. (12  5i)  (4  3i)
8  8i
8. (9  2i)  (2  5i)
7  7i
9. (15  12i)  (11  13i)
26  25i 26–29 3
24. (2i)(6i)(4i) 48i 25. 3i(5i)2 75i 26. i13 i
10. i4 11. i6 12. i15 30–33, 46, 6 27. i24 1 28. i38 1 29. i63 i
1 1 i 47
Solve each equation. 34–37, 42, 7 30. (5  2i)  (4  4i) 31. (3  5i)  (3  5i) 6 32. (3  4i)  (1  4i) 2
43
13. 5x2  45  0
3i
14. 4x2  24  0
i 6

15. 9x2  9
i 38–41, 44, 8
33. (7  4i)  (3  i) 34. (3  4i)(3  4i) 25 35. (1  4i)(2  i)
45 4i 2 6
Gl

Skills
NAME
/M G

Practice,
______________________________________________
Hill

p. 289 and
287 DATE ____________
GlPERIOD
Al _____
b 2
36. (6  2i)(1  i) 8  4i 37. (3  i)(2  2i) 8  4i 38.    i
5-9 Practice (Average)
48–55 4 3i 5 5
Practice, p. 290 (shown) 56–61 5 4 10 6 10  i 39 14 2i 2 1
Complex Numbers
39.    i 40.    i 41.    i
Simplify. 5  3i 17 17 4  i 17 17 3  4i 5 5
1. 49
 7i  12i 3
2. 612  3. 121
 s8 11s 4i Extra Practice ★ 42. (5  2i)(6  i)(4  3i) 163  16i ★ 43. (2  i)(1  2i)(3  4i) 20  15i
4. 36a
 3b4 5. 8
  32
 6. 15
  25
 See page 839.
6| a| b2i a
 16 515


★ 44. 5  i3 11 53 ★ 45. 1  i2 1 22

7. (3i)(4i)(5i) 8. (7i)2(6i) 9. i 42
46. (i  4)x2      i    i
60i 294i 1
5  i3
14 14 3 3 1  i2
10. i 55 11. i 89 12. (5  2i)  (13  8i) (3  i )x  2  4i
i i 8  10i
46. Find the sum of ix2  (2  3i)x  2 and 4x2  (5  2i)x  4i.
13. (7  6i)  (9  11i) 14. (12  48i)  (15  21i) 15. (10  15i)  (48  30i)
16  5i 3  69i 38  45i More About . . . 47. Simplify [(3  i)x2  ix  4  i]  [(2  3i)x2  (1  2i)x  3].
16. (28  4i)  (10  30i) 17. (6  4i)(6  4i) 18. (8  11i)(8  11i)
18  26i 52 57  176i (5  2i )x2  (1  i )x  7  i
6  5i 5  6i
19. (4  3i)(2  5i) 20. (7  2i)(9  6i) 21.  
2i 2 Solve each equation.
23  14i 75  24i
2
22.  
7  8i
113
14  16i 3i
23.   52i
7i 2  4i
24.  1  i
1  3i
48. 5x2  5  0 i 49. 4x2  64  0 4i

Solve each equation.

50. 2x2 
 12  0 i 6 
51. 6x2  72  0 2i 3
25. 5n2  35  0 i 7
 26. 2m2  10  0 i 5
 52. 3x2 
 9  0 i 3 53. 2x2  80  0 2i 10 
27. 4m2  76  0 i 19

29. 5m2  65  0 i 13

28. 2m2  6  0 i 3
3
30.  x2  12  0 4i
 2
3i 5
54. x2  30  0
3
 4 
55. x2  1  0 
5
5 i
4 2
67 19
Find the values of m and n that make each equation true. Find the values of m and n that make each equation true. 61. , 
31. 15  28i  3m  4ni 5, 7 32. (6  m)  3ni  12  27i 18, 9
11 11
33. (3m  4)  (3  n)i  16  3i 4, 6 34. (7  n)  (4m  10)i  3  6i 1, 4
56. 8  15i  2m  3ni 4, 5 57. (m  1)  3ni  5  9i 4, 3
7 5
35. ELECTRICITY The impedance in one part of a series circuit is 1  3j ohms and the
Electrical 58. (2m  5)  (1  n)i  2  4i , 3 59. (4  n)  (3m  7)i  8  2i , 4
impedance in another part of the circuit is 7  5j ohms. Add these complex numbers to 2 3
find the total impedance in the circuit. 8  2j ohms

36. ELECTRICITY Using the formula E  IZ, find the voltage E in a circuit when the
Engineering ★ 60. (m  2n)  (2m  n)i  5  5i 3, 1 ★ 61. (2m  3n)i  (m  4n)  13  7i
current I is 3  j amps and the impedance Z is 3  2j ohms. 11  3j volts
The chips and circuits in
Gl NAME
/M G ______________________________________________
Hill 290 DATE ____________
Gl PERIOD
Al _____
b 2
computers are designed by 62. ELECTRICITY The impedance in one part of a series circuit is 3  4j ohms, and
Reading
Readingto to Learn
5-9
Mathematics,
Learn MathematicsELL
p. 291 electrical engineers. the impedance in another part of the circuit is 2  6j. Add these complex
Complex Numbers
numbers to find the total impedance in the circuit. 5  2j ohms
Pre-Activity How do complex numbers apply to polynomial equations?
Read the introduction to Lesson 5-9 at the top of page 270 in your textbook.
Online Research
Suppose the number i is defined such that i 2  1. Complete each equation.
2 4
To learn more about ELECTRICAL ENGINEERING For Exercises 63 and 64, use the formula E  I  Z.
2i 2  (2i)2  i4  1
electrical engineering,
63. The current in a circuit is 2  5j amps, and the impedance is 4  j ohms. What
Reading the Lesson visit: www.algebra2.
1. Complete each statement. is the voltage? 13  18j volts
com/careers
a. The form a  bi is called the

b. In the complex number 4  5i, the real part is

standard form
4
of a complex number.

and the imaginary part is 5 .

64. The voltage in a circuit is 14  8j volts, and the impedance is 2  3j ohms. What
This is an example of a complex number that is also a(n) imaginary number. is the current? 4  2j amps
c. In the complex number 3, the real part is 3 and the imaginary part is 0 .

This is example of complex number that is also a(n) real number. 274 Chapter 5 Polynomials
d. In the complex number 7i, the real part is 0 and the imaginary part is 7 .

This is an example of a complex number that is also a(n) pure imaginary number.

2. Give the complex conjugate of each number.

NAME ______________________________________________ DATE ____________ PERIOD _____
a. 3  7i 3  7i
b. 2  i 2i Enrichment,
5-9 Enrichment p. 292
3. Why are complex conjugates used in dividing complex numbers? The product of
complex conjugates is always a real number.
Conjugates and Absolute Value
When studying complex numbers, it is often convenient to represent a complex
number by a single variable. For example, we might let z  x  yi. We denote
4. Explain how you would use complex conjugates to find (3  7i)  (2  i). Write the the conjugate of z by z. Thus, z  x  yi.
division in fraction form. Then multiply numerator and denominator by
2  i. We can define the absolute value of a complex number as follows.

 z   x  yi  
x2  y2

Helping You Remember There are many important relationships involving conjugates and absolute
values of complex numbers.
1  3

5. How can you use what you know about simplifying an expression such as  to
2  5

help you remember how to simplify fractions with imaginary numbers in the Example 1 Show  z 2  zz
 for any complex number z.
denominator? Sample answer: In both cases, you can multiply the
numerator and denominator by the conjugate of the denominator. Let z  x  yi. Then,
z  (x  yi)(x  yi)
 x2  y2
  (x2  y2 )2
  z2

274 Chapter 5 Polynomials

 65. CRITICAL THINKING Show that the order relation “” does not make sense for
the set of complex numbers. (Hint: Consider the two cases i 0 and i  0. In About the Exercises…
each case, multiply each side by i.) See pp. 283A–283B.
Organization by Objective
66. WRITING IN MATH Answer the question that was posed at the beginning of
• Add and Subtract Complex
the lesson. See margin. Numbers: 18–33, 46–61
How do complex numbers apply to polynomial equations? • Multiply and Divide
Include the following in your answer: Complex Numbers: 34–45
• how the a and c must be related if the equation ax2  c  0 has complex Odd/Even Assignments
solutions, and Exercises 18–61 are structured
• the solutions of the equation 2x2  2  0. so that students practice the
same concepts whether they
Standardized 67. If i2  1, then what is the value of i71? C are assigned odd or even
Test Practice A 1 B 0 C i D i problems.
68. The area of the square is 16 square units. What is the
area of the circle? C Assignment Guide
A 2 units2 B 12 units2 Basic: 19–41 odd, 47–59 odd,
C 4 units2 D 16 units2 63, 65–68, 71–85
Average: 19–63 odd, 65–68,
Extending PATTERN OF POWERS OF i 69. 1, i, 1, i, 1, i, 1, i, 1
the Lesson 71–85 (optional: 69, 70)
69. Find the simplified forms of i6, i7, i8, i9, i10, i11, i12, i13, and i14.
70. Explain how to use the exponent to determine the simplified form of any
Advanced: 18–64 even, 65–85
power of i. See margin.

Maintain Your Skills

Mixed Review Solve each equation. (Lesson 5-8)
4 Assess
71.  72.  73. 
x  5  x  5 4
3
2x  1  5 12 x  3  1  3 11 Open-Ended Assessment
Speaking Have students discuss
Simplify each expression. (Lesson 5-7) 1
 the meaning and “reality” of
75. y 
7
1
 2  1 2
  1 3
 a 4
74. x 5
x x
3 15 2 3 y 3
76. a 4
 imaginary numbers, including
a
their graphical representation
For Exercises 77–80, triangle ABC is reflected over y
and their usefulness in electrical
1 2
 
the x-axis. (Lesson 4-6) A
77. 2 engineering.
3 2 1 77. Write a vertex matrix for the triangle. C


78. 1 0
0 1  78. Write the reflection matrix. O
x
79. Write the vertex matrix for A’B’C’.
79. 2 1 2
  80. Graph A’B’C’. See pp. 283A–283B.
B Intervention
3 2 1
New Suggest that
students who
81. FURNITURE A new sofa, love seat, and coffee table cost $2050. The sofa costs are confused
twice as much as the love seat. The sofa and the coffee table together cost $1450.
How much does each piece of furniture cost? (Lesson 3-5) sofa: $1200, love by imaginary
seat: $600, coffee table: $250 numbers think of i as a very
Graph each system of inequalities. (Lesson 3-3) 82–83. See pp. 283A–283B. special kind of variable that
82. y  x  1 83. x  y  1 most of the time can be treated
y 2x  2 x  2y  4 similar to the variable x.

Find the slope of the line that passes through each pair of points. (Lesson 2-3)
1
84. (2, 1), (8, 2)  85. (4, 3), (5, 3) 0 Assessment Options
10
www.algebra2.com/self_check_quiz Lesson 5-9 Complex Numbers 275 Quiz (Lessons 5-8 and 5-9) is
available on p. 308 of the Chapter 5
Resource Masters.
Answers
66. Some polynomial equations have 70. Examine the remainder when the
complex solutions. Answers should exponent is divided by 4. If the
include the following. remainder is 0, the result is 1. If the
• a and c must have the same sign. remainder is 1, the result is i. If the
remainder is 2, the result is 1. And if
• i
the remainder is 3, the result is i.

Lesson 5-9 Complex Numbers 275

 Study Guide
and Review
Vocabulary and Concept Check
Vocabulary and
absolute value (p. 272) dimensional analysis (p. 225) polynomial (p. 229) scientific notation (p. 225)
Concept Check binomial (p. 229) extraneous solution (p. 263) power (p. 222) simplify (p. 222)
coefficient (p. 222) FOIL method (p. 230) principal root (p. 246) square root (p. 245)
• This alphabetical list of complex conjugates (p. 273) imaginary unit (p. 270) pure imaginary number (p. 270) standard notation (p. 225)
vocabulary terms in Chapter 5 complex number (p. 271) like radical expressions (p. 252) radical equation (p. 263) synthetic division (p. 234)
includes a page reference conjugates (p. 253) like terms (p. 229) radical inequality (p. 264) terms (p. 229)
where each term was constant (p. 222) monomial (p. 222) rationalizing the denominator trinomial (p. 229)
introduced. degree (p. 222) nth root (p. 245) (p. 251)

• Assessment A vocabulary Choose a word or term from the list above that best completes each statement or
test/review for Chapter 5 is phrase.
available on p. 306 of the 1. A number is expressed in when it is in the form a
10n, where
Chapter 5 Resource Masters. 1  a  10 and n is an integer. scientific notation
2. A shortcut method known as is used to divide polynomials by
binomials. synthetic division
Lesson-by-Lesson 3. The is used to multiply two binomials. FOIL method
Review 4. A(n) is an expression that is a number, a variable, or the product of
a number and one or more variables. monomial
For each lesson, 5. A solution of a transformed equation that is not a solution of the original
equation is a(n) . extraneous solution
• the main ideas are 6. are imaginary numbers of the form a  bi and a  bi. Complex conjugates
summarized, 7. For any number a and b, if a2  b, then a is a(n) of b. square root
• additional examples review 8. A polynomial with three terms is known as a(n) . trinomial
concepts, and 9. When a number has more than one real root, the is the
• practice exercises are provided. nonnegative root. principal root
10. i is called the . imaginary unit

Vocabulary
PuzzleMaker
ELL The Vocabulary PuzzleMaker 5-1 Monomials
software improves students’ mathematics See pages Concept Summary
vocabulary using four puzzle formats— 222–228.
crossword, scramble, word search using a • The properties of powers for real numbers a and b and integers m and n
are as follows.
word list, and word search using clues.
1
Students can work on a computer screen an  n , a  0 (am)n  amn
a
or from a printed handout.
am  an  am  n (ab)m  ambm
am n

an
 am  n, a  0 ban  ban , b  0
MindJogger • Use scientific notation to represent very large or very small numbers.
Videoquizzes
Examples 1 Simplify (3x4y6)(8x3y).
ELL MindJogger Videoquizzes (3x4y6)(8x3y)  (3)(8)x4  3y6  1 Commutative Property and products of powers
provide an alternative review of concepts  24x7y7 Simplify.
presented in this chapter. Students work
in teams in a game show format to gain 276 Chapter 5 Polynomials www.algebra2.com/vocabulary_review
points for correct answers. The questions
are presented in three rounds.
TM

Round 1 Concepts (5 questions) Ask students to review their Foldable and make sure that their
Round 2 Skills (4 questions) notes, diagrams, and examples are complete. Since journal entries
Round 3 Problem Solving (4 questions) are personal, remind students that these journals are shared only
For more information with their consent. Ask if anyone would like to describe one of
about Foldables, see their journal entries, perhaps something they had difficulty with but
Teaching Mathematics later cleared up by asking questions.
with Foldables. Encourage students to refer to their Foldables while completing
the Study Guide and Review and to use them in preparing for the
Chapter Test.

276 Chapter 5 Polynomials

 Chapter 5 Study Guide and Review Study Guide and Review

2 Express each number in scientific notation.

a. 31,000 b. 0.007
31,000  3.1
10,000 0.007  7
0.001
 3.1
104 10,000  104  7
103 0.001  1 1
 or 
1000 103

Exercises Simplify. Assume that no variable equals 0.

See Examples 1–4 on pages 222–224.
1 2 16
13. (2y)(4xy3) 8xy4 14. c2fcd c4d 2f
3 4
11. f 7  f 4 3 12. (3x2)3 27x6
f 5 3 15

Evaluate. Express the result in scientific notation. See Examples 5–7 on page 225.
5,400,000
15. (2000)(85,000) 1.7  108 16. (0.0014)2 1.96  106 17.  9  102
6000

5-2 Polynomials
See pages Concept Summary
229–232.
• Add or subtract polynomials by combining like terms.
• Multiply polynomials by using the Distributive Property.
• Multiply binomials by using the FOIL method.
Examples 1 Simplify (5x2  4x)  (3x2  6x  7). 2 Find (9k  4)(7k  6).
5x2  4x  (3x2  6x  7) (9k  4)(7k  6)
 5x2  4x  3x2  6x  7  (9k)(7k)  (9k)(6)  (4)(7k)  (4)(6)
 (5x2  3x2)  (4x  6x)  7  63k2  54k  28k  24
 2x2  2x  7  63k2  26k  24
18. 3c  1 19. 4x2  22x  34 20. 18m3n  78m3  30m2n
Exercises Simplify. See Examples 2–5 on pages 229 and 230.
18. (4c  5)  (c  11)  (6c  17) 19. (11x2  13x  15)  (7x2  9x  19)
20. 6m (3mn  13m  5n)
2 21. x8y10(x11y9  x10y6) x 3y  x2y4
22. (d  5)(d  3) 23. (2a  6)2
2 24. (2b  3c)3
d 2  2d  15 4a4  24a2  36 8b3  36b2c  54bc2  27c3

5-3 Dividing Polynomials

See pages Concept Summary
233–238.
• Use the division algorithm or synthetic division to divide polynomials.
Example Use synthetic division to find (4x4  x3  19x2  11x  2)
(x  2).
2 4 1 19 11 2
8 14 10 2
4 7 5 1 0 → The quotient is 4x3  7x2  5x  1.
26. 10x3  5x2  9x  9 3
Exercises Simplify. See Examples 1–5 on pages 233–235. 25. 2x3  x  
x3
25. (2x4  6x3  x2  3x  3)  (x  3) 26. (10x4  5x3  4x2  9)  (x  1)
27. (x2  5x  4)  (x  1) x  4 28. (5x4  18x3  10x2  3x)  (x2  3x)
5x2  3x  1
Chapter 5 Study Guide and Review 277

Chapter 5 Study Guide and Review 277

Study Guide and Review Chapter 5 Study Guide and Review

5-4 Factoring Polynomials

See pages Concept Summary
239–244.
• You can factor polynomials using the GCF, grouping, or formulas
involving squares and cubes.

Examples 1 Factor 4x3  6x2  10x  15.

4x3  6x2  10x  15  (4x3  6x2)  (10x  15) Group to find the GCF.
 2x2(2x  3)  5(2x  3) Factor the GCF of each binomial.
 (2x2  5)(2x  3) Distributive Property

2 Factor 3m2  m  4.
Find two numbers whose product is 3(4) or 12, and whose sum is 1. The two
numbers must be 4 and 3 because 4(3)  12 and 4  (3)  1.
3m2  m  4  3m2  4m  3m  4
 (3m2  4m)  (3m  4)
 m(3m  4)  (1)(3m  4)
 (3m  4)(m  1)

Exercises Factor completely. If the polynomial is not factorable, write prime.

See Examples 1–3 on pages 239 and 241. 30. 2(5a2  1)(a  2) 31. (5w2  3)(w  4)
29. 200x2  50 50(2x  1)(2x  1) 30. 
10a3  2a  4
20a2
31. 5w3  20w2  3w  12 32. x4  7x3  12x2 x2(x  3)(x  4)
33. s3  512 (s  8)(s2  8s  64) 34. x2  7x  5 prime

5-5 Roots of Real Numbers

See pages Concept Summary
245–249.
n n
Real n th roots of b, b, or b
n n n
n b if b 0 b if b 0 b if b  0
one positive root
even no real roots
one negative root
one real root, 0
one positive root no positive roots
odd
no negative roots one negative root
7
Examples 1 Simplify 
81x6. 2 Simplify  
2187x1
4y35.

2187x14y35 

81x6   
2187x14
y35  
(3x2y5
7 7
(9x3)2 81x6 = (9x3)2 )7 (3x2y5)7
 9x3 Use absolute value.  3x2y5 Evaluate.

Exercises Simplify. See Examples 1 and 2 on pages 246 and 247.

 16
35. 256  6
36. 216 37.  38. 
3 5
(8)2 8 c5d15 cd 3
39. 
(x4   40. 
(512  41.  42. 
a2  1 
3 4
3)2 x2)3 16m8 2m2 0a  25
x4  3 512  x2 a  5
278 Chapter 5 Polynomials

278 Chapter 5 Polynomials

 Chapter 5 Study Guide and Review Study Guide and Review

5-6 Radical Expressions

See pages Concept Summary
250–256.
For any real numbers a and b and any integer n 1,
• Product Property:  ab  a  b
n n n

a

n
n a
• Quotient Property: b  
b
n

5 5
Example Simplify 6 2.
32m3  51024m

6 2  6  5

32m3  51024m (32m3 
 1024
5 5 5
m2) Product Property of Radicals

 30
25  45
5
 m5 Factor into exponents of 5 if possible.

 30
25   
45  
5 5 5
m5 Product Property of Radicals

 30  2  4  m or 240m Write the fifth roots.


Exercises Simplify. See Examples 1–6 on pages 250–253. 47. 20  86
6
 22
43. 128  35
44. 5  20  53
  375
45. 512
6

5
  811
46. 611
5
 47. 8  12
211  2
5
48. 8  15
  21 
 670
 9
243 1 3  5  
10   5
210
49.  50.   51. 
3 3  
5 4 4  2 7

5-7 Radical Exponents

See pages Concept Summary
257–262.
• For any nonzero real number b, and any integers m and n, with n 1,
m
b n  bm  b 
n n m

4 2 3x
Examples 1 Write 32 5  32 5 in radical form. 2 Simplify  3 .
z
4 2 4  2 3x 3x
32 5  32 5  32 5 5
Product of powers   Rational exponents
z
3 1

6 z3
 32 5 Add. 2

6 3x z3 Rationalize the
 (25) 5 32  25 1

 2 denominator.
z3 z3
 26 or 64 Power of a power
2

3xz 3 3x z Rewrite in
3 2
  or  radical form.
z z

Exercises Evaluate. See Examples 3 and 5 on pages 258 and 259.

 1  9
2 1 5 2
54. 8 3 
 
52. 27 3  53. 9 3  9 3 81
9 27 4
Simplify. 3 See Example 5 on page 259. 2
 
1 y5 xy xyz 3 3x  4x 2 5 8
55.   56.   57.   
3x 3  4x 3
z z 2
2 3
y5 y
 3
x

Chapter 5 Study Guide and Review 279

Chapter 5 Study Guide and Review 279

 • Extra Practice, see pages 836–839.
Study Guide and Review • Mixed Problem Solving, see page 866.

5-8 Radical Equations and Inequalities

See pages Concept Summary
263–267.
• To solve a radical equation, isolate the radical. Then raise each side of the
equation to a power equal to the index of the radical.

Example Solve 
3x  8  1  3.
3x  8  1  3 Original equation

 3x  8  2 Subtract 1 from each side.

3x
  8   22
2
Square each side.

3x  8  4 Evaluate the squares.

x4 Solve for x.

Exercises Solve each equation. See Examples 1–3 on pages 263 and 264.
1 3
58. x  6 36 59. y 3  7  0 343 60. (x  2) 2  8 no solution
61. 
x530 4 62. 
3t  5  3  4 18 63. 
2x  1  3 5

64. 2x 65. y 
 5  2y 
 3 8 66. y 
 1  y
4
 1  2 8.5 45 8

5-9 Complex Numbers

See pages
Concept Summary
270–275.

• i2  1 and i  1
• Complex conjugates can be used to simplify quotients of complex
numbers.

Examples 1 Simplify (15  2i)  (11  5i).

(15  2i)  (11  5i)  [15  (11)]  (2  5)i Group the real and imaginary parts.
 4  3i Add.

7i
2 Simplify  .
2  3i
7i 7i 2  3i
     2  3i and 2  3i are conjugates.
2  3i 2  3i 2  3 i
14i  21i2
  Multiply.
4  9i
2

21  14i 21 14
  or   i i 2  1
13 13 13

Exercises Simplify. See Examples 1–3 and 6–8 on pages 270, 272, and 273. 68. 10  10i
12
67. 64m  8m6i 68. (7  4i)  (3  6i) 69. 69   24 72
70. i 6 1 71. (3  4i)(5  2i) 23  14i 72. 6  i6  i 7
1i 4  3i 2 11 3  9i 3  21i
73.  i 74.    i 75.  
1i 1  2i 5 5 4  2i 10

280 Chapter 5 Polynomials

280 Chapter 5 Polynomials

 Practice Test

Vocabulary and Concepts

Assessment Options
Choose the term that best describes the shaded part of each trinomial.
Vocabulary Test A vocabulary
1. 2 x2  3x  4 c a. degree test/review for Chapter 5 can be
2. 4x 2  6x  3 a b. constant term found on p. 306 of the Chapter 5
c. coefficient Resource Masters.
3. 9x2  2x  7 b
Chapter Tests There are six
Skills and Applications Chapter 5 Tests and an Open-
Ended Assessment task available
Simplify. 6. 8h3  72h2  216h  216 in the Chapter 5 Resource Masters.
4. (5b)4(6c)2 22,500b4c2 5. (13x  1)(x  3) 13x2  38x  3 6. (2h  6)3

Evaluate. Express the result in scientific notation. Chapter 5 Tests

7,200,000  0.0011
7. (3.16
103)(24
102) 7.584  106 8. 
0.018
4.4  105 Form Type Level Pages
Simplify. 1 MC basic 293–294
9. (x4  x3  10x2  4x  24)  (x  2) 10. (2x3  9x2  2x  7)  (x  2) 31 2A MC average 295–296
x3  x2  8x  12 2x2  5x  12   2B MC average 297–298
Factor completely. If the polynomial is not factorable, write prime. x2
11. x2  14x  45 (x  5)(x  9) 12. 2r2  3pr  2p2 (2r  p)(r  2p) 13. x2  23x  3 x  3 2 2C FR average 299–300
2D FR average 301–302
Simplify.
3 FR advanced 303–304
 57
14. 175  16. 36  554
15. 5  3 7  23  29  33  186

45  93
18. 9 2  9 3  6 3 18
1 2 1 7
9  1 7 1
17.  
  
19. 11 2  11 3  11 6 1331 MC = multiple-choice questions

5  3 22
4 1 FR = free-response questions

6 v 11
b 1 2

t18 2st 3 

7
20. 
256s11

21. v 11 
6
4s 5 22.  
v 3

b b
2
1

2 b  1 Open-Ended Assessment
Solve each equation. Performance tasks for Chapter 5
23. 
b  15     
24. 2x 25. 
4
3b  1 7 x  4 no solution y  2  9  14 623 can be found on p. 305 of the
26.    11  18 172 27. 
4x  28  
6x  38 5 28. 1  
x  5  
3
2w  1 x  12 4 Chapter 5 Resource Masters. A
Simplify. sample scoring rubric for these
29. (5  2i)  (8  11i) 3  9i 30. (14  5i)2 171  140i
tasks appears on p. A34.

31. SKYDIVING The approximate time t in seconds that it takes an object to fall
a distance of d feet is given by t  d
. Suppose a parachutist falls 11 seconds
16
TestCheck and
Worksheet Builder
before the parachute opens. How far does the parachutist fall during this time
period? 1936 ft This networkable software has
32. GEOMETRY The area of a triangle with sides of length a, b, and c is given
three modules for assessment.
by 
s(s  a
)(s  b
1
)(s  c), where s  (a  b  c). If the lengths of the sides • Worksheet Builder to make
2 worksheets and tests.
of a triangle are 6, 9, and 12 feet, what is the area of the triangle expressed in
radical form?

 2
2715 • Student Module to take tests
ft
33. STANDARDIZED TEST PRACTICE 2  x    D
1 2 4 on-screen.
x
1 1 • Management System to keep
A 2 B 4 C x2  2 D x2  2  4
x x student records.
www.algebra2.com/chapter_test Chapter 5 Practice Test 281

Portfolio Suggestion
Introduction In this chapter, you have divided and simplified
monomials, polynomials, radical expressions, and complex numbers, often
using procedures that involved a series of steps.
Ask Students Write a description for your portfolio comparing these
various division problems. Identify which type of division problems was
most challenging for you and explain why you think this is true. Be sure to
include several examples of your work from this chapter.

Chapter 5 Practice Test 281

 Standardized
Test Practice

These two pages contain practice 6. In rectangle ABCD, AD is 8 units long.
Part 1 Multiple Choice B
What is the length of A  in units? C
questions in the various formats
that can be found on the most Record your answers on the answer sheet A 4 A B
frequently given standardized provided by your teacher or on a sheet of
paper. B 8
tests.
1. If x3  30 and x is a real number, then x lies
C 83
30˚
A practice answer sheet for these between which two consecutive integers? B D 16 D C
two pages can be found on p. A1 A 2 and 3

of the Chapter 5 Resource Masters. B 3 and 4

NAME DATE PERIOD C 4 and 5 7. The sum of two positive consecutive integers
Standardized
5 Standardized Test Practice
Test Practice is s. In terms of s, what is the value of the
Student Recording
Student Record Sheet,
Sheet (Use with pages 282–283 of p. A1Edition.)
the Student
D 5 and 6 greater integer? D
Part 1 Multiple Choice
2. If 12x  7y  19 and 4x  y  3, then what is s s1
Select the best answer from the choices given and fill in the corresponding oval. A   1 B 
1 A B C D 4 A B C D 7 A B C D the value of 8x  8y? C 2 2
s s1
2 A B C D 5 A B C D 8 A B C D
A 2 C  D 
3 A B C D 6 A B C D 9 A B C D
2 2
Part 2 Short Response/Grid In
B 8
Solve the problem and write your answer in the blank.
C 16
Also enter your answer by writing each number or symbol in a box. Then fill in
the corresponding oval for that number or symbol.
8. Latha, Renee, and Cindy scored a total of
10 12 14 16
D 22 30 goals for their soccer team this season.
/ / / / / / / /
Latha scored three times as many goals as
.

1
.
0
1
.
0
1
.
0
1
.

1
.
0
1
.
0
1
.
0
1
.

1
.
0
1
.
0
1
.
0
1
.

1
.
0
1
.
0
1
.
0
1
3. For all positive integers n, Renee. The combined number of goals
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
scored by Latha and Cindy is four times the
n  n  1, if n is even and
3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5

number scored by Renee. How many goals

5 5 5 5 5 5 5 5
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
7 7 7 7 7 7 7 7

1
7 7 7 7 7 7 7 7

n  (n  1), if n is odd.

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
did Latha score? C
11 13 15 17 2
/ / / / / / / /
What is 8
13 ? B A 5 B 6
. . . . . . . . . . . . . . . .

1
0
1
0
1
0
1 1
0
1
0
1
0
1 1
0
1
0
1
0
1 1
0
1
0
1
0
1 A 42
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
C 18 D 20
5
6
7
5
6
7
5
6
7
5
6
7
5
6
7
5
6
7
5
6
7
5
6
7
5
6
7
5
6
7
5
6
7
5
6
7
5
6
7
5
6
7
5
6
7
5
6
7
B 49
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

C 56
Part 3 Quantitative Comparison
9. If s  t  1 and t  1, then which of the
Answers

Select the best answer from the choices given and fill in the corresponding oval. D 82
18 A B C D 20 A B C D 22 A B C D following must be equal to s2  t2? D
19 A B C D 21 A B C D
4. Let x ❅ y  xy – y for all integers x and y. A (s  t)2 B t2  1
If x ❅ y  0 and y  0, what must x equal? D
A 2
C s2 1 D st

Additional Practice B 1
See pp. 311–312 in the Chapter 5 C 0
Resource Masters for additional D 1
standardized test practice. Test-Taking Tip
5. The sum of a number and its square is three
Question 9 If you simplify an expression and
times the number. What is the number? D
do not find your answer among the given answer
A 0 only
choices, follow these steps. First, check your answer.
B 2 only Then, compare your answer with each of the given
answer choices to determine whether it is
C 2 only equivalent to any of the answer choices.
D 0 or 2
282 Chapter 5 Standardized Test Practice

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.

282 Chapter 5 Polynomials

 Aligned and
verified by

Part 3: QUANTITATIVE COMPARISON

Part 2 Short Response/Grid In Part 3 Quantitative Comparison
Record your answers on the answer sheet Compare the quantity in Column A and the
provided by your teacher or on a sheet of quantity in Column B. Then determine
paper. whether:
1
10. Let a ✿ b  a  , where b  0. What is the A the quantity in Column A is greater,
b
value of 3 ✿ 4? 3.25 or 13/4 B the quantity in Column B is greater,
C the two quantities are equal, or
11. If 3x2  27, what is the value of 3x4? 243
D the relationship cannot be determined
from the information given.
12. In the figure, if x  25 and z  50, what is
the value of y? 105 Column A Column B

x˚ 18. s and t are positive integers.

z˚ s
t s
s st
y˚
A

13. For all positive integers n, let n equal the

19. The original price of a VCR is discounted
greatest prime number that is a divisor of n. by 20%, giving a sale price of $108.
70 the original price
What does equal? 7/3 $130
27 of the VCR
A
5y
14. If 3x  2y  36 and   5, then x  ? .4
3x
20.
15. In the figure, a square with side of length w
w
2 2 is inscribed in a circle. If the area of the
circle is k, what is the exact value of k? 4 3w

the area of the area of

the rectangle the circle
B

21. k and n are integers.

16. For all nonnegative numbers n, let n be kn  64
defined by n    . If n  4, what is
n
2 k n
the value of n? 64 D

17. For the numbers a, b, and c, the average

(arithmetic mean) is twice the median. If 22. For all positive integers m and p, let
a  0, and a  b  c, what is the value m ✢ p  4(m  p)  mp.
c
of ? 5 8✢3 3✢8
b
C
www.algebra2.com/standardized_test Chapter 5 Standardized Test Practice 283

Chapter 5 Standardized Test Practice 283

 Pages 231–232, Lesson 5-2 2
44. 2x 3  x 2  1  
3. x x x 3x  1
45. x  3
2 2 2
x x x x 3x  7
46. x 2  1  
x 2
2

x x x 47. x  2
2 48. x  3
x x x
59. Division of polynomials can be used to solve for
55. The expression for how much an amount of money will unknown quantities in geometric formulas that apply to
grow to is a polynomial in terms of the interest rate. manufacturing situations. Answers should include the
Answers should include the following. following.
• If an amount A grows by r percent for n years, the • 8x in. by 4x  s in.
amount will be A(1  r)n after n years. When this • The area of a rectangle is equal to the length times
expression is expanded, a polynomial results. the width. That is, A  w.
• 13,872(1  r)3, 13,872r 3  41,616r 2  41,616r  • Substitute 32x 2  x for A, 8x for , and 4x  s for w.
13,872 Solving for s involves dividing 32x 2  x by 8x.
• Evaluate one of the expressions when r  0.04. For A  w
example, 13,872(1  r)3  13,872(1.04)3 or 32x 2  x  8x(4x  s)
$15,604.11 to the nearest cent. The value given in
32x 2  x
the table is $15,604 rounded to the nearest dollar.   4x  s
8x
Additional Answers for Chapter 5

1
Pages 237–238, Lesson 5-3 4x    4x  s
8
21. b 2  10b 1
 s
8
22. x  15
1
23. n2  2n  3 The seam is  inch.
8
6
24. 2c 2  c  5  
c2 Page 243, Lesson 5-4
39 56. Factoring can be used to find possible dimensions of a
25. x 3  5x 2  11x  22  
x2 geometric figure, given the area. Answers should
26. 6w 4  12w 3  24w 2  30w  60 include the following.
27. x2 • Since the area of the rectangle is the product of its
28. x 2  3x  9 length and its width, the length and width are factors
29. y2  y  1 of the area. One set of possible dimensions is
4x  2 by x  3.
30. m2  7
• The complete factorization of the area is
3
31. a3  6a 2  7a  7   2(2x  1)(x  3), so the factor of 2 could be placed
a1
5
with either 2x  1 or x  3 when assigning the
32. 2m3  m 2  3m  1   dimensions.
m3
56
33. x 4  3x 3  2x 2  6x  19  
x3
13
34. 3c 4  c 3  2c 2  4c  9  
c2
35. g  5
4
36. 2b 2  b  1  
b1
37. t 4  2t 3  4t 2  5t  10
38. y 4  2y 3  4y 2  8y  16
39. 3t 2  2t  3
51
40. h2  4h  17  
2h  3
2
41. 3d 2  2d  3  
3d  2
42. x 2  x  1
6
43. x 3  x  
2x  3

283A Chapter 5 Additional Answers

Page 262, Lesson 5-7 Page 275, Lesson 5-9
71. The equation that determines the size of the region 65. Case 1: i 0
around a planet where the planet’s gravity is stronger Multiply each side by i to get i 2 0  i or 1 0.
than the Sun’s can be written in terms of a fractional This is a contradiction.
exponent. Answers should include the following. Case 2: i  0
• The radical form of the equation is r  D 
 
5
Mp 2
 or
Since you are assuming i is negative in this case,
you must change the inequality symbol when you


MS
Mp2 multiply each side by i . The result is again
rD 5  . Multiply the fraction under the radical i 2 0  i or 1 0, a contradiction.
MS2
M3
S Since both possible cases result in contradictions,
by 3. the order relation “” cannot be applied to the
MS


complex numbers.
Mp2 Mp3
rD 5
  80. y
MS2 MS3


 D 55
Mp2MS3
MS
B'

5 O x

Mp2MS3 C'
D
5 5

MS A'
5
D
Mp2MS3
 

Additional Answers for Chapter 5

MS 82. y
5
D
Mp2MS3
The simplified radical form is  .
MS y  2x  2
• If Mp and MS are constant, then r increases as D
O x
increases because r is a linear function of D with
positive slope.
yx1
Page 267, Lesson 5-8
44. If a company’s cost and number of units manufactured 83. y
are related by an equation involving radicals or rational
exponents, then the production level associated with a
given cost can be found by solving a radical equation. xy1
Answers should include the following.
3 O x
• C  10 n 2  1500 x  2y  4
2

• 10,000  10n 3  1500 C  10,000
2

8500  10n 3
Subtract 1500 from each side.
2

850  n 3
Divide each side by 10.
3
 3
850 2  n Raise each side to the  power.
2
24,781.55  n Use a calculator.
Round down so that the cost does not exceed $10,000.
The company can make at most 24,781 chips.

Chapter 5 Additional Answers 283B