Copyright

by

Donald Wayne Beaver

2010
 The Report Committee for Donald Wayne Beaver
Certifies that this is the approved version of the following report:

Exploring Methods for Finding Solutions to Polynomial Equations

APPROVED BY
SUPERVISING COMMITTEE:

Supervisor:
Efraim Armendariz

Mark Daniels
Exploring Methods For Finding Solutions to Polynomial Equations

by

Donald Wayne Beaver, B.A.

Report
Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

Master of Arts

The University of Texas at Austin

August 2010
 Dedication

I dedicate this report to my wife of twenty eight years, Gayla Beaver, who has

supported and encouraged me to be the person I am today. Without her help, I would

never have attempted the Masters program at the University of Texas and for that I am

grateful. Throughout our married life Gayla has insisted that I persist in my endeavors

and studies to improve my knowledge and teaching. I thank her for that in this dedication.
 Acknowledgements

I would like to acknowledge Drs. Efraim Armendariz and Mark Daniels for their

support and guidance throughout this Masters program. Their hard work and dedication

to the teachers of today has been an inspiration for all of us in the past three years.

August 2010

v
 Abstract

Exploring Methods for Finding Solutions to Polynomial Equations

Donald Wayne Beaver, M.A.

The University of Texas at Austin, 2010

Supervisor: Efraim Armendariz

There are many methods for solving polynomial equations. Dating back to the

Greek and Babylonian mathematicians, these methods have been explored throughout the

centuries. The introduction of the Cartesian Coordinate Plane by Rene Descartes greatly

enhanced the understanding of what the solutions actually represent.

The invention of the graphing calculator has been a tremendous aid in the

teaching of solutions of polynomial equations. Students are able to visualize what these

solutions represent graphically. This report explores these methods and their uses.

vi
 Table of Contents
List of Tables ...................................................................................................... viii

List of Figures ....................................................................................................... ix

Chapter 1: Introduction ...........................................................................................1

Chapter 2: A Geometrical Approach .......................................................................7

Carlyle's Method ...........................................................................................10
Chapter 3: Iteration as a Technique .......................................................................13
Definition of a Magnification Factor ............................................................14
Fixed Point Iteration Theorem ......................................................................14
Chapter 4: Finding Roots of Higher Degree Polynomials .....................................16
Luddhar's Theorem .......................................................................................21

Chapter 5: Conclusion............................................................................................31

References .............................................................................................................32

Vita .......................................................................................................................33

vii
 List of Tables

Table 1: Number and Nature of Possible Roots..............................................17

Table 2: Results of dividing (1.1) by quadratic factors with rational imaginary

zeros. .................................................................................................20

viii
 List of Figures

Figure 1: Graph of f ( x)  c( x  a ) 2  b ...........................................................5

Figure 2: Graph of f (x ) shifted 3b units down..................................................5

Figure 3: Euclidean Constructions for i - ii ........................................................8

Figure 4: Euclidean Constructions for iii - iv.....................................................8

Figure 5: Graph of ( x  1) 2  ( y  1) 2  5 ........................................................11

Figure 6: y1  2 x  3; y 2  x 2 ........................................................................12

Figure 7: f ( x)  x 2  2ax  a 2  b 2 ................................................................23

Figure 8: f ( x)  ( x  a )( x 2  2bx  b 2  c 2 ) ...................................................23

Figure 9: Graphs of f ( x)  x 3  8; g ( x)  3x  6 ............................................25

Figure 10: Graph of f ( x)  x 2  x  1 . ..............................................................26

Figure 11: Graph of f ( x)  x 2  x  1 . ..............................................................26

Figure 12: g ( x )  x 2  x  1 ..............................................................................28

Figure 13: f ( x )  x 2  x  1 using Modulus Surfaces......................................29

Figure 14: f ( x )  x 2  x  1 .............................................................................29

Figure 15: Paths of zeros of x 2  x  a 0 as a 0 varies over  1,1 .....................29

ix
 Chapter 1: Introduction

Typical secondary curricula introduce the basic concept of functions with first

order linear equations. Teachers invest great time, effort, and creativity as they convey

the ideas of obtaining solutions and the meanings behind those solutions. In solving

linear equations, the concept of balancing an equation can be made concrete and tangible

by allowing students to experience the phenomenon with a balance scale. There exists a

seemingly endless supply of manipulatives which students can use to visualize most

fundamental mathematics concepts. However, making the transition from solving these

basic first order equations to higher order polynomials can prove quite challenging for

both student and teacher. It is at this point in mathematics when one must move from the

concrete to the abstract. Whereas the balance scale provides a core foundational

understanding of solving linear equations, the zero product theorem is not so easily

demonstrated. While the basic idea of the theorem itself seems attainable, using this as a

method to solve quadratic equations is not always readily understood by students.

Additionally, because so much emphasis is placed on finding the one solution to a first

order equation, students often have great difficulty grasping the concept that more than

one solution may exist to a given equation.

The introduction of the graphing calculator has been a tremendous aid in allowing

the student to actually see the points where the function crosses the x-axis.

Unfortunately, making the connection from the viewing window to the written

polynomial equation requires students make the abstract connection between the zero

product theorem and x-intercepts. While solving a linear equation can be done without

much mathematical depth, solving quadratics demands students have a fairly in depth

understanding of what it means to be a function, the relationship between the equation

1
and its domain and range, and how the function itself produces the points associated with

the graph. The students are now able to really connect the solutions obtained by the

quadratic formula, completing the square, and factoring to what they see in the viewing

window. Fortunately, once a student reaches this level of understanding, the transition to

cubic, quartic, and higher degree equations proves far less taxing.

Once certain concepts are understood by the student, the x-intercepts are easily

seen as the inputs into a polynomial function that generate the number 0 as the output, or

solutions of the polynomial equation P(x) = 0. Specifically these concepts are:

i. what it means to be a function

ii. the meaning of Domain and Range of a function
iii. all inputs of the Domain generate outputs in the Range that
are associated with points on the graph of the function

Similarly, cubic, quartic, and higher degree equations are seen in the same

manner. The methods used to find the solution to quadratic equations, such as completing

the square, factoring, and the quadratic formula, are taught in first year Algebra classes,

but a connection (or meaning) comes to fruition when the actual graphs are seen.

Another problem arises when solutions to quadratics are complex numbers that

are not real numbers. It is hard for the student to visualize what is taking place since the

complex solutions are not often seen or are not easy to graph. The quadratic formula

yields these complex solutions, but what is their meaning? And when discussing the
solutions to a cubic equation, how do these solutions appear in the graph of the

polynomial associated with that equation?

Because most of what is discussed in secondary mathematics education in relation

to polynomial functions and their graphs is “local behavior,” finding the zeros of a

polynomial is crucial to the understanding of the function. The previously discussed

2
methods for finding solutions and what they mean are subjects addressed in this report.

Even though each of these methods is of very little value in themselves, when used in

combination with each other, a function’s behavior can be discovered. Further use of

methods learned in calculus such as the derivative of a function or relative extrema, are

also helpful but not discussed here.

Polynomials of a second degree, or quadratics, take three forms, each of which

has its own purpose. The three forms and their use are:

Name Form Use

Standard Form f ( x)  ax 2  bx  c easy to generate points,
especially the y-intercept

Vertex Form f ( x)  c( x  a) 2  b easy to find vertex (a, b)

and graph

Root Form f ( x)  a( x  r1 )( x  r2 ) easy to find the x-intercepts

(r1, 0) and (r2, 0)

When graphing the parabola generated by a quadratic, the vertex form is

preferred. Once the student has manipulated the equation into this form, a simple chart

with the vertex in the center entry and two values of x to the left and right (or

above/below the y in the case of horizontal parabolas) is sufficient to see its shape.

If the graph of the parabola intersects the x-axis, these are referred to as the real

zeros of the function. In the case of parabolas that do not cross the x-axis, the numbers

that can be substituted for x that generate the number 0 for y are still complex solutions.

Consider a parabola whose equation in vertex form is

f ( x)  c( x  a ) 2  b .

3
As mentioned earlier, the vertex is (a, b). Letting f(x) = 0 and solving for x,

0 = c(x – a)2 + b

b
  (x – a)2
c

b
  =x–a
c

b
x= a   .
c

Depending on the algebraic signs of b and c, the parabola will have real or

complex roots, but the spacing about the axis of symmetry is the same. In Figure 1, the
roots are   i , where  the x-coordinate, or abscissa of the vertex, and  is half the

length of the chord determined by the horizontal line y = 2b and where b is the y-
coordinate, or ordinate, of the vertex. The chord is referred to as the latus rectum and

passes through the focus of the parabola. As the parabola is shifted down, these become

real roots and the horizontal line is simply the x-axis, as illustrated in the transition from

Figure 1 to Figure 2 below.

4
 Figure 1. Graph of f ( x)  c( x  a ) 2  b [6, p. 248]

Figure 2. Graph of f (x) shifted 3b units down [2, p. 248]

A simple example would be to consider the parabola with vertex (0, -1) and c = 1

f ( x)  x 2  1

5
and with real roots at (  1, 0) . If the parabola was shifted up two units to have a new

vertex at (1, 0), but c remains the same, the new complex roots are (  i, 0) .

6
 Chapter 2: A Geometrical Approach

Long before Rene Descartes introduced the Cartesian Coordinate Plane, the Greek

mathematicians found solutions to quadratic equations geometrically. A few of these

methods will be discussed here.

A well known theorem in mathematics resulting from triangle similarity states

that in a right triangle the altitude drawn to the hypotenuse is the geometric mean
a b
between the segments into which it is divided. In the proportion  , b is the
b c

geometric mean of a and c.

Simple geometric constructions can be made using a straight edge and compass,

as shown in Figure 3. Then some use of simple algebra and geometry provides visual

solutions that are generated for the following quadratic equations:

i. x 2  bx  c  0

ii. x 2  bx  c  0

iii. x 2  bx  c  0

iv. x 2  bx  c  0

7
Figure 3. Euclidean constructions for i. and ii. [4, p. 363]

Figure 4. Euclidean Constructions for iii. and iv. [4, p. 363]

8
The first equation (i.) can be verified by setting x1  AC and x2  BC . The previously

mentioned theorem then states

AC c

c BC

or

c  AC  BC .

But since

x1  AC and BC  b  x1 ,

it follows that

x1 (b  x1 )  c

bx1  x 2  c

x 2  bx1  c  0.

9
 Similarly the second equation ii. can be solved. However, equations iii. and iv.
make use of another well known theorem stating that the square of a tangent segment
drawn from a point in the exterior of a circle is equal to the product of the secant segment
and the outer secant segment.

Carlyle’s Method.
In the early 1800’s another method was suggested using the intersection of a
circle with the x-axis to show solutions to the equation x 2  bx  c  0 . For if the

equation of a particular circle with a diameter having endpoints at (0, 1) and (-b,c) given
by x 2  y 2  bx  (1  c) y  c  0 is evaluated for y = 0, the equation simplifies to the

desired quadratic.
To illustrate, consider the quadratic equation

x 2  2x  3  0 .

If the circle whose diameter has endpoints at (0, 1) and (-2, -3) is graphed, the roots of
the quadratic equation are also the abscissas of the points of intersection of the circle with
the x-axis as shown in Figure 5. In this case x1  3 and x 2  1 .

10
 y

(0,1)

(-3,0) (1,0)
x

(-1,-1)

(-2,-3)

Figure 5. Graph of ( x  1) 2  ( y  1) 2  5

It is customary to solve a quadratic equation by graphing its corresponding

quadratic function as a parabola and locating points where the graph crosses the x-axis.
Hornsby suggests an alternate method which incorporates the graph of the “parent
function” y = x2 and somewhat simplifies the process. [3, p.364] Solving the quadratic
equation

x 2  bx  c  0

is equivalent to finding the points of intersection of the line y  bx  c . This

phenomenon is easily seen because if the left side of the equation is solved for x2, the
equation becomes
x 2  bx  c.

11
In the example used earlier, y = x2 + 2x – 3, the points of interest are the intersection of
y1 = -2x + 3 with y2 = x2 , i.e. (-3, 9) and (1, 1) as shown in Figure 6 below:

(-3,9)

(1,1)

Figure 6. y1  2 x  3; y 2  x 2

The abscissas of these points are the solutions to the original quadratic equation.

12
 Chapter 3: Iteration as a Technique

An alternate method of solving polynomial equations is offered by Butts with

some interesting results. [2] The process will be demonstrated here with a quadratic even
though use of the Quadratic Formula would be an easier method of solution. Consider the
function

P( x)  x 2  2 x  2 .

The goal is to find roots of the polynomial, that is the x-values that make P(x) = 0 or, the
x-values that satisfy the equation

x 2  2x  2  0 .

Instead of proceeding with the Rational Root Theorem, Butts suggests putting the
equation in an alternate form and applying iteration to this equation to find a fixed point.
A fixed point, x0 , is a particular value of x such that f ( x 0 )  x 0 .

One alternate form could be

x2  2
x .
2

Iteration is the process by which a seed (or initial) value x0 is substituted into the right
side of this equation, then the result f ( x 0 ) is then substituted for x until the two values

are identical, the aforementioned fixed point. For if the output (functional value) is equal
to the input (x-value) accurate to any desired decimal approximation, that particular value
13
satisfies the alternate form and therefore will satisfy the original form, yielding the
desired result. If a seed value converges to a fixed point, the root has been found.
However, sometimes the seed value will diverge to   in which case iteration fails. To
ensure convergence, the following definition and ensuing theorem guarantee intervals of
convergence.

Definition. The magnification factor MF of f(x) at x = xo is

f ( xo   )  f ( xo   )
MF ( f ( xo )) = lim .
 0 2

This definition leads us to the theorem: The Fixed Point Iteration Algorithm converges if
MF ( f ( x))  1 for values of x near the seed value x0 " [2, p. 5]. In this example,

f ( xo   )  f ( xo   )
| MF ( f ( xo )) |  lim
 0 2

 ( xo   ) 2  2   ( xo   ) 2  2 
  
 lim  2   2 
 0 2

x o  2xo   2  2  x o  2xo   2  2
2 2

 lim
 0 2

4xo
 lim
 0 4

 xo ,
14
so that convergence is guaranteed if xo  1, i.e.,  1  xo  1.

The theorem doesn’t necessarily determine the smallest interval of convergence,

merely one that guarantees convergence. Choosing the seed value to be 0, it can be
verified after 42 iterations with a scientific calculator that the sequence converges to
approximately 0.732051, a root of the polynomial accurate to six decimal places. If a
seed value of 3 is chosen, the sequence diverges to  and no root is found.
There exist three types of fixed points; attracting, repelling, and neutral. The
iteration method only finds roots when a fixed point is an attracting fixed point.

15
 Chapter 4: Finding Roots Of Higher Degree Polynomials

Finding the roots of higher degree polynomials is much more difficult than
finding the roots of linear or quadratic functions. A few theorems and properties make the
process easier.
i. If r is a root of a polynomial function, then ( x  r ) is a factor of the

polynomial,
ii. Any polynomial function with real coefficients can be written as the
product of linear factors ( x  r ) and quadratic factors (ax 2  bx  c)

which are irreducible over the real numbers.

A quadratic factor that is irreducible over the reals is a quadratic function with no
real zeros; equivalently those that have a negative discriminant.
Another helpful theorem is Descartes’ Rule of Signs, which states that the number
of variations in (algebraic) signs throughout the polynomial determines the number of
positive roots that the function will have. The Rule will not tell where the polynomial’s
roots are, but will tell how many to expect when finding them. Consider the following
polynomial in its original form:

f ( x)  2 x 5  x 4  2 x 3  4 x 2  x  3.

Without concern for the actual values of the coefficients themselves, notice that the
algebraic signs change four times. Thus there will be, at most, four positive roots.
Conveniently, the roots come in “pairs,” so if one positive root is found, finding another
positive root is in order.

16
To find the number of negative roots, f(-x) is generated:

f ( x)  2( x) 5  ( x) 4  2( x) 3  4( x) 2  ( x)  3

f ( x )  2 x 5  x 4  2 x 3  4 x 2  x  3 .

Since there is only one sign change, this polynomial has exactly one negative root. So
once that root is found, looking for another negative root becomes moot. Therefore, there
are 4, 2, or 0 positive roots and exactly 1 negative root. To illustrate the “nature of the
roots,” that is, the number and type (real or complex), a chart is generated.

Table 1. Number and nature of possible roots

Number of positive Number of negative Number of

real roots real roots complex roots

4 1 0

2 1 2

0 1 4

The Rational Root Theorem also can be used as an aid to finding roots of a
polynomial function of the form:

f ( x )  a n x n  a n 1 x n 1  ...  a1 x  a 0 .

17
Assuming the coefficients are all integers and a root of the polynomial is rational, the
numerator of the root is always a factor of a0 and the denominator is a factor of an. In the
example above, if any rational roots exist, they must come from the “bank” of

1 3
 1,  3,  , or  .
2 2

As stated earlier, each of these theorems is not all that helpful alone, but when
used in combination with one another, finding roots becomes easier. The idea is to use
these tools in combination in order to find a rational root. This is usually done by
synthetic division, then compressing the equation and repeating the process until the
polynomial is expressed as the product of linear factors or irreducible quadratic factors
that can be solved using the quadratic formula. Students sometimes find this process
laborious and tedious. If, however, a student realizes some important facts about the
process, there comes a realization that even though a root is not found by synthetic
division, it is not a waste of time. The remainder from synthetic division is synonymous
with the functional value, so it still yields a point on the graph. Also, if f (0) is positive
1
but f (1) is negative, a logical choice to try next would be ; since the polynomial is a
2

function its graph must cross the x-axis between 0 and 1. That is, the graph cannot “go
around” that portion because then it would not be a function. Of course, there is always
the possibility that the root may be irrational.
Taking the Rational Root Theorem to a higher level, Redmond [7] proves a
theorem that states “If P(x) is a polynomial in the form

f ( x )  a n x n  a n 1 x n 1  ...  a1 x  a 0 n2

18
where a0 , an, and f (1) are all three odd numbers, then f (x) has no rational roots.” [7]
While this does not apply to the previous function (a n  2) , let’s consider the example

Redmond gives:

f ( x)  x 5  7 x 4  28 x 3  125 x 2  x  3275

in which a5 = 1, a 0 = -3275, and f (1) = -3169, all odd numbers. Redmond assures that

this particular polynomial has no rational roots. An easier example, one which will be
discussed later, is

f ( x)  x 2  x  1

1 i 3
a2 = 1, a0 = 1, and f(1) = 3 (all odd numbers) which has as its roots { },
2

obviously complex roots.

As seen earlier, the Rational Root Theorem simply narrows the search for rational
roots of the polynomial. In many cases a great majority of roots can be eliminated before
attempting the tedious process of synthetic division. Barrs, J. Braselton, and L. Braselton
offer the Imaginary Rational Root Theorem, which narrows the search for imaginary
roots. [1] The theorem states:

P ( x )  a n x n  a n 1 x n 1  ...  a1 x  a 0

19
 p q
is an nth degree polynomial function with integer coefficients. If x     i   i
r r
are rational imaginary zeros of P(x), where  and   0 are rational, p, q, and r are

integers, then r2 is a divisor of an and p2 + q2 is a divisor of a0.

To illustrate this theorem, consider the following polynomial function:

P( x)  x 4  2 x 3  6 x 2  2 x  5 . (1.1)

The possible rational roots are 1, -1, 5, and -5. Synthetically dividing with these possible
roots yields remainders of 16, 8, 1040, and 520, respectively, therefore there are no
rational roots. An application of the imaginary rational root theorem indicates that since
1 = 02 + 12 and 5 = 12 + 22, the possible complex rational roots would be given by
1  2i,  1  2i, and  i with corresponding factors x 2  2 x  5, x 2  2 x  5, and x2 + 1.

The chart below gives the results when P(x) is divided by each of these potential factors.

Table 2. Results of dividing (1.1) by quadratic factors with rational complex zeros.

Possible Zero Possible Factor Quotient Remainder

1 2i x 2  2x  5 x 2  4x  9 -40

 1 2i x 2  2x  5 x2 1 0

i x2 1 x 2  2x  5 0

Therefore,
x 4  2 x 3  6 x 2  2 x  5  ( x 2  1)( x 2  2 x  5)

and

20
 x 4  2 x 3  6 x 2  2 x  5 = 0 if x  i or x  1  2i .

Another interesting (but perhaps not so useful) theorem is afforded by Luthar. [5]
According to Luthar, Luddhar’s Theorem states “If P ( x)  ax 3  bx 2  cx  d , with a, b,
c, and d integers, a  0, b  0, the function has a rational root if there exist non-zero
lm pq
integers l, m, p, and q such that c = l + m, b = p + q,  p, and  l . That rational
d a
l
root is given by - [5, p. 107]. The following example illustrates Luddhar’s Theorem.
p

Let

P ( x)  6 x 3  2 x 2  15 x  5 .

6 15
Since  , the function can be written as
2 5

P( x)  2 x 2 (3x  1)  5(3x  1)

= (3x  1)(2 x 2  5)

1
which yields a rational root of - . The theorem and its process essentially amount to
3

“factoring by grouping” for this example, but the next problem illustrates its worth.
Consider a polynomial that is not factorable by grouping:

P( x)  x 3  6 x 2  11x  6 .

21
Select two integers whose sum is 11, the coefficient of the term involving x. For example,
let l = 9 and m = 2. Now p and q must be determined. So

lm 9  2
p=  3
d 6

so that

q = b – p = 6 – 3 = 3.

Checking the other condition on q, namely,

la 9  1
q=   3;
p 3

thus Luddhars’s conditions are satisfied. Consequently, the original function can now be
written as

P ( x)  x 3  3x 2  3x 2  9 x  2 x  6

 x 2 ( x  3)  3x( x  3)  2( x  3)

 ( x  3)( x 2  3x  2)

 ( x  3)( x  1)( x  2) .

22
 Consider a quadratic function with complex roots given by a  bi . Using a well

known fact that the quadratic function with roots r1 and r2 is given by

f ( x)  x 2  (r1  r2 ) x  (r1  r2 )

 x 2  [(a  bi )  (a  bi )]x  [(a  bi )(a  bi )]

 x 2  2ax  a 2  b 2 .

This equation can be easily transformed by completing the square into

y  b 2  ( x  a) 2

from which it is evident that, if OA and AP are measured as shown in Figure 7. The
complex roots will be given by a  ib  OA  i AP .

Figure 7. f ( x)  x 2  2ax  a 2  b 2 Figure 8. f ( x)  ( x  a )( x 2  2bx  b 2  c 2 )

23
 To illustrate, consider the following quadratic function:

f ( x)  x 2  2 x  4

or
f ( x )  x 2  2  1  x  12  ( 3 ) 2 ,

where a  1 , b  3. It is easily shown by completing the square that the function can be

written in vertex form as

f ( x)  ( x  1) 2  3 .

The vertex of this parabola is (1, 3) and OA = 1 while AP = 3. According to Yanosik [8],
the complex roots would be given by

OA  i AP  1  i 3 .

Using the quadratic formula to generate the roots, this fact is seen to be true. Similarly,
the cubic function given by

f ( x)  ( x  a )( x 2  2bx  b 2  c 2 )

will yield the real root a and the complex roots b  ic by the following method. As

shown in Figure 8, a line is drawn through A(a, 0) tangent to the graph at P. BP is

measured and the slope m of the tangent line is calculated. Yanosik suggests (and proves
in another article) the complex roots are

24
 b  ic  BP  i m .

An example to illustrate follows. The roots of the aforementioned quadratic will

help in the illustration. Take the cubic function given by:

f ( x)  x 3  8

 ( x  2)( x 2  2 x  4)

The graphs of this function and the corresponding tangent are shown in Figure 9.

Figure 9. Graphs of f ( x)  x 3  8; g ( x)  3x  6

From the graph (and the factored form of the function), it can be seen that the real root is
–2. If Yanosik is correct, the complex roots are given by

b  ic  BP  i m ,

indicating that BP = 1 and the slope of the tangent is 3. The line shown tangent in Figure
9 indeed has a slope of 3 and is tangent at the point (1, 9) yielding BP = 1.
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 In a previously mentioned polynomial function, the real roots of the polynomial
seem to “disappear” as the graph is shifted up. For example, Figure 10 shows a graph of
the following function:
f ( x)  x 2  x  1

Figure 10. Graph of f ( x)  x 2  x  1 .

Figure 11. Graph of f ( x)  x 2  x  1 .

1 5
Notice that the function has real roots at x = . However when the function with
2

equation

26
 f ( x)  x 2  x  1

is graphed (see Figure 11), the roots become imaginary.

Complex numbers are defined to be expressions of the form a  bi treated as
residues to the modulus i 2  1 . Modulus in this sense is used in the same way as modulus

in a clock. That is, a time of 20 o’clock is considered to be 8 o’clock (mod 12). It’s the
residue, or remainder, when 20 is divided by 12. In using the Rational Root Theorem
mentioned earlier, when a possible root is “checked” by synthetic division, if the
remainder is zero, the number is a root. However if there is a remainder other than zero,
that remainder is the same as the functional value at that value of x. Therefore, it is a
point on the graph of the polynomial.
Similarly, this phenomenon occurs with polynomial residues to the modulus
y 2  1 . If any polynomial in y (with real coefficients) is divided by y 2  1 the remainder
is of the form a  by with a and b being real numbers. The polynomial mentioned before

P ( x)  x 2  x  1

has no real solutions, but

x 2  x  1  0 (mod y 2  1)

1 y 3 1 y 3
has solutions x  because if x  then
2 2

2
 1 y 3   1 y 3 
x  x  1  
2
 
 
 1

 2   2 
27
 1  2 y 3  3y 2  2  2 y 3 4
  
4 4 4

1  2 y 3  3y 2  2  2 y 3  4

4

3  3y 2

4

3
 (1  y 2 )  0 (mod y 2  1).
4
Long and Hern use modulus surfaces to explain this phenomenon. In Figure 10,
the graph shows the real roots mentioned earlier. [4] Figure 12 shows taking the absolute
value of the function, gives somewhat of an insight as to the appearance of the modulus
surface associated with this function as seen in Figure 13. The “low points” of the surface
signify the real roots. If the function is shifted up two units, the surface is that shown in

Figure 14. The appearance of the two are similar, but seem to be rotated horizontally
2

radians.

Figure 12. g ( x)  x 2  x  1

28
 Figure 13. f ( x )  x 2  x  1 Figure 14. f ( x )  x 2  x  1

Figure 15. Paths of zeros of x 2  x  a 0 as a 0 varies over  1,1 .

The zeros of a polynomial are continuous functions of the coefficients of that

polynomial. Hence as the coefficient a0 of the polynomial z2 + z + a0 changes
continuously from –1 to +1 on the real axis, the positions of the low points on the
corresponding modulus surfaces move along a continuous path along the complex plane

29
with the beginning and end positions as shown in Figure 15. Since the zeros are given by
 1  1  4a 0 1
, they will remain real until a 0  at which point they become complex.
2 4
1 1
When a 0  , the polynomial z 2  z  is a perfect square and has a real zero at
4 4
1
x   . Considering cubic functions of the nature f ( x )  x 3  a 0 , as a 0 varies over an
2
interval [-  ,  ] , it is seen that the roots appear to be at the endpoints of the “spokes of a

wheel,” with a rotation of radians, depending on whether  is positive or negative as
3

shown in Figure 16. This phenomenon is consistent with the concepts illustrated by
DeMoivre’s Theorem.

Figure 16. Paths of zeros near a triple zero of x 3  a 0 for a 0 in   ,  

30
 Chapter 5: Conclusion

Several methods for solving polynomial equations have been introduced in this
report. Depending on the nature of the polynomial and what is known, some methods are
more appropriate than others. The underlying fact is that the more methods that are
known, the more proficient one becomes at solving these equations.
Most secondary mathematics teachers agree that the introduction of the graphing
calculator has greatly aided in student learning. While some might argue that the
understanding of algorithms and processes is essential to advancement in mathematics,
the introduction of the graphing calculator into lessons can assist in enhancing the
understanding of these concepts.

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 References

1. Sharon Barrs, James Braselton, Lorraine Braselton, A Rational Root Theorem for
Imaginary Roots, The College Mathematics Journal 34, No. 5, (2003) 280-382.

2. Thomas Butts, Fixed Point Iteration – An Interesting Way to Begin a Calculus Course,
The Two-Year College Mathematics Journal 12 (1991) 2-7.

3. E John Hornsby, Jr. Geometrical and Graphical Solutions of Quadratic Equations, The
College Mathematics Journal 21 (1990) 362-369.

4. Cliff Long and Thomas Hern, Graphing the Complex Zeros of Polynomials Using
Modulus Surfaces, The College Mathematics Journal 20 (1989) 98-105.

5. R.S. Luthar, Luddhar’s Method of Solving a Cubic Equation with a Rational Root, The
Two-Year College Mathematics Journal 11 (1980) 107-110.

6. Alec Norton and Benjamin Lotto, Complex Roots Made Visible¸The College
Mathematics Journal (1984) 248-249.

7. Don Redmond, Finding Rational Roots of Polynomials, The College Mathematics

Journal 20 (1989) 139-141.

8. George Yanosik, Graphical Solutions for Complex Roots of Quadratics, Cubics, and
Quartics, National Mathematics Magazine 17 (1943) 147-150.

32
 Vita

Don Beaver received a Bachelor of Arts in Mathematics from the University of

Texas at Austin in 1972 and a Master of Arts in Mathematics in 2010. He currently
resides in Austin, Texas and teaches high school mathematics. He has been married to
Gayla Beaver for twenty eight years and has two daughters, Alyssa and Cara.

Email address: dabeav345@aol.com

This report was typed by Don Beaver.

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