5 Polynomial Functions...2011/08/30 · Approximations and Graphing p. 253 5.4 It’s Fundamental The Fundamental Theorem of Algebra p. 259 5.5 When Division Is Synthetic Polynomial

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5.1 Many Terms

Introduction to Polynomial Expressions,

Equations, and Functions ● p. 237

5.2 Roots and Zeros

Solving Polynomial Equations and

Inequalities: Factoring ● p. 245

5.3 Successive Approximations, Tabling,

Zooming/Tracing, and Calculating

Solving Polynomial Equations:

Approximations and Graphing ● p. 253

5.4 It’s Fundamental

The Fundamental Theorem

of Algebra ● p. 259

5.5 When Division Is Synthetic

Polynomial and Synthetic Division ● p. 263

5.6 Remains of a Polynomial

The Remainder and Factor

Theorems ● p. 273

5.7 Out There and In Between

Extrema and End Behavior ● p. 279

Fish tanks are often described by their volume, or how many gallons of water they hold. Many

fish tanks are rectangular prisms, which are three-dimensional figures formed by six rectangular

faces that meet at right angles. You will use polynomial functions to model the volume of

various rectangular prisms.

5 C HA PT E R

Polynomial Functions

Chapter 5 ● Polynomial Functions 235

236 Chapter 5 ● Polynomial Functions

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Lesson 5.1 ● Introduction to Polynomial Expressions, Equations, and Functions 237

5

ObjectivesIn this lesson, you will:

● Identify polynomial expressions and

functions.

● Write polynomial expressions in

ascending and descending order.

● Recognize a polynomial written in

standard form.

● Add, subtract, and multiply polynomials.

● Graph polynomial functions.

Key Terms● cubic

● polynomial function

● standard form of a polynomial

function

● degree of a polynomial

● polynomial expression

● polynomial equation

● polynomial inequality

● continuous function

● zeros of a polynomial function

5.1 Many TermsIntroduction to Polynomial Expressions,Equations, and Functions

Problem 1 Building a BoxYou plan on building some plywood boxes in the form of rectangular prisms.

For each box, the width will be 6 inches less than the height, and the length will be

24 inches more than the height.

1. Define a variable for the height, and write expressions for the width and length.

2. Using this variable, define a function for the volume and another for the surface area.


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3. Calculate the volume and surface area if the height of one box is

a. 10 inches

b. 20 inches

c. 40 inches

4. The expression that represents the volume is called a cubic or third-degree

expression, and the expression that represents the surface area is a quadratic

or second-degree expression. Graph these functions using a graphing

calculator and sketch the resulting graphs on the grid shown. Use �60 � x � 60

and �1,000,000 � y � 3,000,000 for the graphing window.

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5. How would you describe the graph of the volume function? The surface area

function?

6. Which portion of these graphs models the problem situation? Why?

7. Determine the zeros for the volume and surface area functions.

8. How many zeros does each have? Why?

9. How does the graph of a cubic function differ from the graph of a quadratic

function? Linear function?

Problem 2 Linear and QuadraticFor each of the following functions, complete a table and graph the function.

1. f(x) � 3x � 4

x y


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2. f(x) � x2 � 3x � 4

x y

3. Determine the domain, the range, the zero(s) (x-intercepts), and the y-intercept(s)

for each of the functions.

a. f(x) � 3x � 4

i. Domain:

ii. Range:

iii. Zero(s):

iv. y-intercept(s):

b. f(x) � x2 � 3x � 4

i. Domain:

ii. Range:

iii. Zero(s):

iv. y-intercept(s):

Both of these functions are examples of a large class of functions named

polynomial functions, or polynomials, meaning “many terms.” These functions are

formed by adding and subtracting terms of the form axb, where a is any real number

and b is a non-negative integer. The standard form of a polynomial function is

f(x) � anxn � an�1xn�1 � . . . � a2x2 � a1x � a0x0. Polynomials in standard form are

written in either ascending form (lowest exponent to highest) or descending form

(highest exponent to lowest). The degree of a polynomial is the largest exponent.

The standard form of a polynomial expression, equation, and inequality is also

written in order of ascending or descending degree.

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Problem 3For each of the following functions, complete a table and graph the function.

1. f(x) � 3x3 � 4x2 � 3x � 4

x y

2. f(x) � x4 � 13x2 � 36

x y


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Polynomial functions as a class are well behaved, meaning that they have smooth,

continuous graphs. The graph of a continuous function is one that can be drawn

without lifting your pencil or has no holes or jumps, as opposed to discontinuous

graphs that jump, skip, or have holes. The zeros of a polynomial function are often

useful and important values for various applications. Calculating these zeros is

helpful in understanding the behavior and graphs of polynomial functions and has

been an emphasis of the study of algebra.

3. Determine the degree, the domain, the range, the zeros (x-intercepts), and the

y-intercept(s) for each of the functions in Questions 1 and 2.

a. f(x) � 3x3 � 4x2 � 3x � 4

i. Degree:

ii. Domain:

iii. Range:

iv. Zeros:

v. y-intercept(s):

b. f(x) � x4 � 13x2 � 36

i. Degree:

ii. Domain:

iii. Range:

iv. Zeros:

v. y-intercept(s):

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For each of the following pairs of polynomial expressions:

a. Write them in descending order.

b. State their degrees.

c. Add them.

d. Subtract the second from the first.

4. 2x3 � 5x2 � 4x5 � 4x; �x3 � 2x4 � 3x � 6

a.

b.

c.

d.

5. x6 � 7x2 � 3x7 � 4; �3x3 � 2x4 � 4x5 � 12

a.

b.

c.

d.

6. �4x2 � 8x3 � 11x; �x3 � 2x4 � 3x � 6

a.

b.

c.

d.

7. 5x � 7x3 � 7x4; �6x � 7x4 � 5

a.

b.

c.

d.


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Multiply the following pairs of polynomial expressions, and write the product in

standard ascending form.

8. �2x3 � 3x4; �3x � x2 � 4

9. �5x2 � 4x; 5x4 � 2x2 � 3x

10. If f(x) � x2 � 2 and g(x) � �4x � 5, calculate the following:

a. f(x) � g(x)

b. f(x) � g(x)

c. f(x) • g(x)

d. f(x) • g(x)

11. If f(x) � 3x � 2 and g(x) � �x2 � 1, calculate the following:

a. f(x) � g(x)

b. f(x) � g(x)

c. f(x) • g(x)

d. f(x) • g(x)

Be prepared to share your work with another pair, group, or the entire class.

Take NoteTo multiply expressions you

can use the multiplication

table or apply the distributive

property.

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Lesson 5.2 ● Solving Polynomial Equations and Inequalities: Factoring 245

5


● Calculate roots of polynomial equations

by factoring.

● Calculate zeros of polynomial functions

by factoring.

Key Term● quartic

5.2 Roots and ZerosSolving Polynomial Equations and Inequalities: Factoring

Problem 1 Polynomial Equations1. Calculate the roots or solutions of the following polynomial equations by

factoring them.

a. x2 � 7x � 10 � 0

b. x3 � 3x2 � x � 3

2. Calculate the zero(s) or x-intercept(s) of the following polynomial functions by

factoring them.

a. f(x) � x2 � 5x � 6


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b. f(x) � x4 � 29x2 � 100

3. What conclusion(s) can you draw about calculating the roots and zeros of

polynomial functions and the solutions of polynomial equations?

Problem 2For each of the following, calculate the solutions or zeros where appropriate.

1. f(x) � x4 � 4x3 � x2 � 4x

2. x4 � 1

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3. f(x) � (2x � 4)(x � 1)(5 � 3x)

As the degree of the polynomial equation or function increases, factoring becomes

more difficult. For quadratics, we were able to derive a formula that enabled us to

solve any quadratic equation. Although there are formulas to solve any cubic or

quartic (fourth-degree) equation, they are complex and reveal the limitations of

straightforward brute-force algebraic manipulations. However, if the functions

are represented in factored form or in a recognizable common form (e.g., difference

of two squares, perfect square trinomial, etc.), calculating solutions or zeros of

higher-order polynomials by factoring is an efficient method.

Problem 3Solving polynomial inequalities is more complex than solving absolute value or

quadratic inequalities. When you solve polynomial inequalities, you need to examine

several different cases.

Let’s start by solving the following quadratic inequality.

1. First factor the quadratic expression.

2. The product of these two factors must be greater than or equal to zero.

Under what conditions is the product of two numbers equal to or greater

than zero?

3. Based on this information, write two separate compound inequalities that must

be true to meet these conditions.

x2 � 6x � 8 � 0


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4. Solve these compound inequalities and graph your solution set on the number

line. Calculate the solution to the original compound inequality by determining

the union of the solution sets. Check your answers by substituting points that

satisfy the solution set and points that do not satisfy the solution set.

5. As the degree of the polynomial inequality increases, the number of cases

will increase. For example, if the product of four numbers is less than zero, how

many different possible combinations or cases would we have to examine?

List them below.

6. Using the same strategies, solve the following inequalities by factoring the

polynomial inequality, writing the inequalities that represent each case, solving

these inequalities, and determining the union of all the solution sets. Graph the

solution set on the number line.

a. x2 � 7x � 10 � 0

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b. 0 � (2x � 4)(x � 1)(5 � 3x)


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c. x4 � 4x3 � x2 � 4x � 0

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Lesson 5.3 ● Solving Polynomial Equations: Approximations and Graphing 253

5

ObjectivesIn this lesson, you will

● Graph polynomial functions using

graphing calculators.

● Calculate approximate roots by graphing,

tracing, and zooming.

Key Terms● approximate root

● zoom

● trace

5.3 Successive Approximations,Tabling, Zooming/Tracing,and CalculatingSolving Polynomial Equations: Approximations and Graphing

Problem 1 Calculating Approximate Solutionsby Evaluation

As we have seen, calculating solutions of higher-order polynomial equations and zeros

of higher-order polynomial functions by factoring is not very efficient. It is time

consuming, and very few polynomials factor over the integers or rational numbers.

In fact, even for quadratic trinomials, only an infinitesimal number are actually

factorable over these sets. How can we solve higher-order polynomial equations?

The answer to this question consumed many centuries of mathematical reasoning and

work by the greatest mathematicians. Many different strategies were developed to

calculate approximate answers. These approximate answers are often

“good enough.”


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For each of the following polynomial equations, evaluate the corresponding

expressions for the given values in the tables.

1. Evaluate the following polynomial function for the values of x in the table.

x3 � 4x2 � x � 1 � 0

Expression

2. Looking at the values from your table, what conclusions can you draw about the

roots of this equation?

3. Using this information, select a value between two of those values, put it into the

table, and evaluate the expression. Can you now better approximate the root?

Explain.

4. Place this value in the table, and repeat the process three more times.

x x3 � 4x2 � x � 1

�5

�2

�1

0

1

2

5

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5. Are we able to use this method to determine an approximate root? Explain.

Problem 2 Approximate Solutions by TablingWhenever a process is tedious or repetitive, technology is an immense timesaver.

One way to solve this problem would be to use a spreadsheet or table routine to

successively evaluate the expression, increasing in small increments until we zoom

in on an accurate approximation.

1. Using your graphing calculator, input the first polynomial expression from Problem 1

as the first function in your list. Set your table to evaluate x3 � 4x2 � x � 1 � 0

from �5 to �2 by 0.1. What can you conclude about the root?

2. Redo the table by selecting the two most appropriate values from the table and

increment a new table by 0.01. What can you conclude about the root?

a. Continue this process until you have found an approximation to three decimal

points. What is this root?

b. Use this process to approximate the other two roots of this equation.


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Problem 3 Approximate Solutions by Tablingand Zooming

Although the tabling process is much more efficient and faster than the first process,

using one of the two powerful features of the graphing calculator can make this

successive approximation method even faster and easier. Understanding the

relationship between roots of polynomial equations and the zeros of polynomial

functions also gives us an advantage. The zoom and trace features let us graph the

function to first approximation, zoom into the approximation, and trace again. By

repeating this process several times, we can “zoom” in on an approximation.

Using the x3 � 4x2 � x � 1 � 0 one more time, set up an appropriate window to

“see” the graph of the function around the zeros and graph the function.

1. Next, trace the function to a point close to the first zero on the left. Record the

x- and y-values of this ordered pair. How do you know that it is close to a zero?

2. Next, zoom in to this portion of your graph, and trace to another point closer to

the zero. Record the x- and y-values of the ordered pair. How do you know this

ordered pair is closer to zero?

3. Continue this process until you are confident that you have found the

approximate value of this first root to three decimal places. Record the x- and

y-values, and explain why this root is accurate to three decimal places.

4. Use this same process to approximate the other two roots of this equation. Then

record the x- and y-values of these roots.

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Problem 4 CalculateAlthough the process of tracing and zooming is much more efficient and faster than

the other methods, there is an additional feature of the graphing calculator that

actually calculates these values. To understand this feature, we’ll use our knowledge

of the relationship between roots of polynomial equations and the zeros of polynomial

functions.

Graph x3 � 4x2 � x � 1 � 0 one more time by inputting the equation in y1, setting up

an appropriate window to “see” the graph of the function around the zeros.

Next, go to the CALCULATE menu and select “zero.”

The graph will appear with the “trace” cursor blinking, and you will be prompted for the

left bound. Move the cursor to a point on the graph just to the left of the left-most zero.

Press ENTER, and you will be prompted for a right bound. Move the cursor to

a point to the right of the zero and press ENTER.

You will then be prompted to calculate a guess.

Press ENTER once more, and an approximation of the zero will be calculated. If this

approximation is not accurate enough, repeat the process.

1. Use this process to approximate each of the three zeros of the function

2. Which of the four methods was the easiest? Why?

x3 � 4x2 � x � 1 � 0.


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Problem 5For each of the following equations, approximate the roots using any of the methods

described in this lesson.

1. 6x3 � 11x2 � 80x � 112 � 0

2. x4 � 3x3 � 2x2 � 3x � 1 � 0


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Lesson 5.4 ● The Fundamental Theorem of Algebra 259

5

Problem 1 Fundamental Theorem of AlgebraThe Fundamental Theorem of Algebra was first proposed in the early 1600s but

would not be proven until almost two centuries later. Both simple and elegant, it

states that any polynomial equation of degree n must have

exactly n complex roots or solutions; also, every polynomial

function of degree n must have exactly n complex zeros.

However, any solution or zero may be a multiple solution or

zero. The proof of this theorem is beyond the content of

this course, but determining all the roots of a polynomial

equation is not. For each of the following polynomial

equations, state the number of roots, and calculate all of the

roots by any method (factoring; successive approximation;

or graphing, tracing, and zooming).

1. x2 � 1 � 0

2. x2 � 1 � 0


● State the Fundamental Theorem

of Algebra.

● Use the Fundamental Theorem of

Algebra to determine roots and zeros

of polynomial equations and functions.

● Determine the number and characteristics

of the roots and zeros of polynomial

equations and functions.

Key Terms● Fundamental Theorem of Algebra

● complex roots

● multiplicity

● double roots

● triple roots

5.4 It’s FundamentalThe Fundamental Theorem of Algebra

Take NoteWhen a root or zero appears

more than once, it is said

to have a multiplicity. For

example, double roots

have a multiplicity of 2,

and triple roots have a

multiplicity of 3.


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3. 7x3 � 5x2 � 6x � 0

4. x4 � 5x2 � 4 � 0

5. x4 � x2 � 3 � 0

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Lesson 5.4 ● The Fundamental Theorem of Algebra 261

5

Problem 2For each of the following polynomial functions, state the number of zeros and

calculate all of the zeros by any method (factoring; successive approximation; or

graphing, tracing, and zooming).

1. f(x) � x4

2. f(x) � x5 � 13x3 � 36x

3. f(x) � x5 � 5x4 � x � 5

4. f(x) � �x3 � 7x2 � 2x � 1

Calculating roots and zeros can be very time consuming; this is especially true

for irrational or complex roots. In the next lessons, we will be concentrating on

identifying and calculating rational roots and zeros.



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Lesson 5.5 ● Polynomial and Synthetic Division 263

5

Problem 1The Fundamental Theorem of Algebra tells us that every polynomial equation

of degree n must have n roots. This means that every polynomial can be

written as the product of n factors of the form ax � b. For example,

2x2 � 3x � 9 � (2x � 3)(x � 3) � 0. Once we have factored the polynomial

completely, we can calculate the roots by setting each factor equal to 0. We also

know that a factor of a number must divide into the number evenly with a remainder

of zero. Factors of polynomials must also divide a polynomial evenly without a

remainder. The division algorithm for dividing one polynomial by another is very

similar to the long division algorithm for whole numbers.


● Divide polynomials.

● Use synthetic division.

Key Term● synthetic division

5.5 When Division Is SyntheticPolynomial and Synthetic Division


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1. This algorithm of Divide–Multiply–Subtract–Bring down–Repeat is familiar and

straightforward. Determine the quotient for each for the following polynomial

division problems.

a. x�4x3 � 0x2 � 7x

2x � 3

x � 3�2x3 � 3x � 9 (Multiply: 3(x � 3) )

2x2 � 6x

3x � 9

3x � 9 (Subtract)

0

31

25�775 (Multiply: 1(25) )

75

25

25 (Subtract)

0

2x 3

x � 3�2x2 � 3x � 9 ( Divide : x�3x ) 2x2 � 6x

3x � 9

3 1

25�775 ( Divide: 2�2 ) 75

25

2x

x � 3�2x2 � 3x � 9 (Bring down)

2x2 � 6x ↓ 3x � 9

3

25�775 (Bring down)

75 ↓ 25

2x

x � 3�2x3 � 3x � 9 (Subtract)

2x2 � 6x

3x

3

25�775 (Subtract)

75

2

2x

x � 3�2x2 � 3x � 9

2x2 � 6x

(Multiply: 2x(x � 3) )25

3

�775

75

(Multiply: 3(25) )

x � 3�2x2 � 3x � 9 ( 2x

Divide: x�2x2 )

25�775 ( 3

Divide: 2�7 )

The following illustrates the division algorithms for both polynomials and whole

numbers.

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b.

c.

d. 2x � 3�4x4 � 0x3 � 5x2 � 7x � 9

x � 1�x3 � 0x2 � 0x � 1

x � 4�x3 � 2x2 � 5x � 16


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e.

2. Answer the following questions about Question 1 parts (a)–(e).

a. What do you notice in parts (a), (c), and (d)? Why was this necessary?

b. When there was a remainder, was the divisor a factor of the dividend? Explain.

c. Describe the remainder when you divide a polynomial by a factor.

3. A whole number like 775 that is divided evenly by a factor like 25, can be

written as a product of factors, 775 � 25(31). When a polynomial

equation or a polynomial function

is divided evenly by a factor,

we can write each as a product of two polynomials:

or

where q(x) is the quotient polynomial

For each of the division problems in Question 1 that had a remainder of 0, rewrite

the dividend as the product of the divisor and the quotient.

9x4 � 3x3 � 4x2 � 7x � 2 �

x3 � 0x2 � 0x � 1 �

4x3 � 0x2 � 7x �

f(x) � (x � r )q(x)

� (x � r ) (bn�1xn�1 � bn�2xn�2 � . . . � b2x2 � b1x � b0)

f(x) � anxn � an�1xn�1 � . . . � a2x2 � a1x � a0 x0

� (x � r ) (bn�1xn�1 � bn�2xn�2 � . . . � b2x2 � b1x � b0)

anxn � an�1xn�1 � . . . � a2x2 � a1x � a0x0

f(x) � anxn � an�1xn�1 � p � a2x2 � a1x � a0x0

anxn � an�1xn�1 � p � a2x2 � a1x � a0x0

3x � 2�9x4 � 3x3 � 4x2 � 7x � 2

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a. When we divide a whole number by a number that is not a factor, we say that

it does not divide evenly and has a remainder. We write the remainder as a

fraction. For instance:

When one polynomial does not divide evenly into another, we can write the

answer in a similar form:

or with functions , where q(x) is the quotient and r(x) is the

remainder. For each of the division problems in Question 1 that had a remainder

other than 0, rewrite the dividend as the product of the divisor and the quotient

plus the remainder.

b. For each of the following, perform the indicated division and write the answer

as a product of the divisor and the quotient plus the remainder.

i. Divide f(x) � 4x4 � 3x3 � 5x2 � 2x � 1 by x � 2

f(x)

x � r� q(x) �

r(x)

x � r

bn�1xn�1 � bn�2xn�2 � p � b2x2 � b1x � b0 �R

x � r

anxn � an�1xn�1 � p � a2x2 � a1x � a0x0

x � r�

101

12� 8

5

12

8

12�101

96

5

4x4 � 0x3 � 5x2 � 7x � 9 �

x3 � 2x2 � 5x � 6 �

f(x) �


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ii. Divide f(x) � 3x4 � 2x3 � 0x2 � 2x � 1 by x � 3

f(x) �

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Problem 2Although dividing polynomials is a straightforward method for determining factors, it

can become very time consuming. In 1809, Poalo Ruffini introduced a shortcut for

long division of a polynomial by a linear factor (x � r ), which is called synthetic

division. Synthetic division makes this division more efficient. It uses only the

coefficients of the terms. The following example compares long division of

polynomials with synthetic division.

0

3x � 9

3x � 9

2x � 3

x � 3�2x2 � 3x � 9

2x2 � 6x

3

3

Bring down the 2

3

Multiply 3 by 2 and place in the next column

3

Add the values in the second column

3

Repeat the process until complete

2 �3 �9

6 9

2 3 0

2 �3 �9

6

2 3

2 �3 �9

6

2

2 �3 �9

↓2

2 �3 �9

6 �9

2 �3 0

Long Division Synthetic Division

The quotient is 2x � 3 and the remainder is 0.

Notice that the opposite sign of r is used and that every power must have a place

holder as in long division.


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1. Here are two examples of synthetic division. For each, perform the following

steps:

i. Write the dividend.

ii. Write the divisor.

iii. Write the quotient.

iv. Write the dividend as the product of the divisor and the quotient plus the

remainder.

a.

2

i.

ii.

iii.

iv.

b.

i.

ii.

iii.

iv.

2. Use synthetic division to perform the following divisions. Write the dividend as

the product of the divisor and the quotient plus the remainder.

a. x � 4�x3 � 2x2 � 5x � 6

2 �4 �4 �3 6

�6 30 �78 243

2 �10 26 �81 249

�3

1 0 �4 �3 6

2 4 0 �6

1 2 0 �3 0

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b. f(x) � 3x3 � 4x2 � 8 divided by x � 2

c.

d.

Calculate


g(x)

r(x)

r(x) � 2x � 1

g(x) � 4x5 � 2x3 � 4x2 � 2x � 11

2x5 � 5x4 � 2x2 � 6x � 7

2x � 3Take NoteSynthetic division works only

for (x � r ) divisors. So you

need to rewrite 2x � 3 to the

equivalent root before

doing the synthetic division.

x �3

2� 0

x �3

2

2x � 3 � 0

x �3

2


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Lesson 5.6 ● The Remainder and Factor Theorems 273

5

Problem 1 Remainder TheoremFrom the Fundamental Theorem of Algebra, we know that any polynomial equation

of degree n has n roots, and with the more efficient method of synthetic division,

calculating them becomes less tedious. We are also able to rewrite any polynomial

equation or function as the product of a divisor and a quotient plus the remainder,

f (x ) � d(x)q(x ) � r (x ). If we are trying to find roots or zeros, however, rewriting the

equation or function as the product of a linear divisor times a polynomial quotient

plus a whole number remainder is more useful. Algebraically, this form is written

f (x ) � (x � r )q(x ) � R.

Rewrite each of the following polynomials as the product of (x � 3) and the quotient

plus the remainder.

1. x3 � 27

2. f (x) � x4 � 2x3 � 3x � 2

f(x) � x4 � 2x3 � 3x � 2

x3 � 27


● Use synthetic substitution.

● Use the Remainder Theorem to evaluate

polynomial equations and functions.

● Use the Factor Theorem to calculate

factors of polynomial equations and

functions.

Key Terms● synthetic substitution

● Remainder Theorem

● Factor Theorem

5.6 Remains of PolynomialThe Remainder and Factor Theorems


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3. Evaluate each of the polynomials in Questions 1 and 2 for x � 3. Compare these

values to your answers to Questions 1 and 2.

4. We can write any polynomial as the product of a linear factor and a quotient

polynomial plus a whole number remainder as follows: f (x) � (x � r )q(x ) � R.

Calculate f (r ). What can you conclude about f (r ) and the whole number

remainder R? Explain.

This result of using synthetic division to evaluate a polynomial function for a

specific value r is called synthetic substitution because f(r) = R, the remainder.

This result also provides the basis for the Remainder Theorem, which states that

when any polynomial equation or function is divided by a linear factor (x � r ), the

remainder is the value of the equation or function when x � r.

5. Determine the remainder of each equation or function by evaluating it at the

given value.

a. 3x6 � 2x3 � 3x � 2 divided by x � 1

b. f (x ) � 3x5 � 4x4 � 2x3 � x2 � 5x � 3 divided by x � 2

c.

d. g(x) � 4x3 � x � 2: 2x � 3

x3 � 2x2 � x � 2

2x � 1

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Problem 2 Factor Theorem1. When you divide a whole number by another whole number and the remainder is

zero, what conclusion can you draw about these two numbers?

2. Using the Remainder Theorem, you divide a polynomial by a linear polynomial.

If the remainder is zero, what can you conclude about these two polynomials?

This result is the Factor Theorem, which states that a polynomial has a linear

polynomial as a factor if and only if the remainder is zero; f (x) has x � r as a factor if

and only if f(r ) � 0.

3. For each of the following, determine if the second polynomial is a factor of the

first.

a. 2x 3 � 3x2 � 3x � 2; x � 1

b. x 4 � 3x 3 � 5x � 2; x � 2

c. 5x 4 � 3x 2 � 2; x � 3

4. For each of the following, determine if the given linear polynomial is a factor of

the polynomial function.

a. f(x) � x7 � 3x � 2; x � 1

b. f(x) � x7 � 3x � 2; x � 1

c. g(x) � 4x 3 � 2x2 � 6x � 5; 2x � 1


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We can also use the Factor Theorem to completely factor higher-order

polynomials:

5. For each of the following, determine if the second polynomial expression is a

factor of the first. Then completely factor the first polynomial expression.

a. x3 � 1; x � 1

b. x 4 � 3x2 � 28; x � 2, x � 2

c. x 4 � x 3 � x2 � x � 2; x � 1

� (x � 1) ( x �1

4�

�15

4i ) ( x �

1

4�

�15

4i )

2x3 � 3x2 � 3x � 2 � (x � 1) (2x2 � x � 2)

x ��b � �b2 � 4ac

2a�

�1 � �(1)2 � 4(2) (2)

2(2)�

�1 � ��15

4� �

1

4�

�15

4 i

2x3 � 3x2 � 3x � 2 � (x � 1) (2x2 � x � 2)

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Using the Fundamental Theorem of Algebra, the Remainder Theorem, and the

Factor Theorem, every polynomial equation or function of degree n can be rewritten

as the product of n linear factors of the form x � r, where r is a complex number.

In future lessons, you will use this information to develop a method to calculate all of

the rational roots or zeros of any polynomial equation or function.

Problem 3 Reversing the Process1. Determine the equation that would have each of the following sets of roots.

a. x � �1, 2, �3

b.

2. After doing Question 3 of Problem 2, a student said that anytime there is one

complex root, it seems that there actually must be two, the root and its

conjugate. Is this student correct? Explain.

3. Determine the equation that would have each of the following sets of roots.

a.

b. x �2

3�

�3

2 i, �2

x � �i, 3

4

x � �2, �5

2


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c.


x � 1� i, �1

2,

3

2

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Lesson 5.7 ● Extrema and End Behavior 279

5


● Graph power functions.

● Determine multiple zeros of polynomial

functions.

● Determine extrema of polynomial

functions.

● Describe the end behavior of polynomial

functions.

Key Terms● power functions

● even function

● odd function

● absolute minimum

● absolute maximum

● relative minimums and maximums

● extremum (extrema)

● end behavior

5.7 Out There and In BetweenExtrema and End Behavior

Problem 1 Power FunctionsIn the last several activities, we have concentrated on the zeros or roots of

polynomial functions and expressions; however, there are a number of

characteristics of these functions that can be important in solving applications and

using graphs to model phenomena. The basic functions of the polynomial functions

are called the power functions.

Notice that for every even-degree power function, such as f(x) � x4, f(x) � f(�x), and

for every odd-degree power function, such as f(x) � x3, f(x) � �f(�x). Functions

that satisfy these conditions are called even and odd functions, respectively. For

example:

● If f(x) � x4, then f(2) � 24 � 16, f(�2) � (�2)4 � 16, and f(2) � f(�2).

● If g(x) � x3, then g(2) � 23 � 8, �g(�2) � �(�2)3 � 8, and g(2) � �g(�2).


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1. Using a graphing calculator, graph each power function. Then sketch its graph on

the grid.

a. f(x) � x

b. f(x) � x2

c. f(x) � x3

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d. f(x) � x4

e. f(x) � x5

f. f(x) � x6


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2. How many x-intercepts does each graph in Question 1 have? What do you notice

about the x-intercept(s)?

3. f(x) � x can be rewritten as a factor of the form (x � 0),

where 0 is the zero of the function. Factor f(x) � x2 into

two factors of the form (x � a), where a is the zero.

4. Factor f(x) � x4 into four factors of the form (x � a),

where a is the zero.

5. In Questions 3 and 4, the functions have multiple zeros that are the same. Look

again at the graphs of the power functions in Question 1, and describe how the

graphs behave at the x-axis based on the number of multiple zeros.

Problem 2 Multiplicity of ZerosIn an earlier activity, you worked with the vertex form of a quadratic function. For

example, in the function f(x) � 2(x � 3)2 � 2, the vertex is (3, �2) and the constant

of dilation is 2. From the factored form of this function, f(x) � 2(x � 4)(x � 2), you

can determine that the zeros are 4 and 2. The second-degree power function is the

basic function for all quadratics.

1. Graph f(x) � 2(x � 3)2 � 2 on the grid.

Take NoteThe Fundamental Theorem

of Algebra states that any

polynomial equation of

degree n has exactly

n complex roots.

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2. Using a graphing calculator, graph each cubic function. Then sketch its graph on

the grid.

a. f(x) � x(x � 2)(x � 2)

b. f(x) � (x � 2)2 (x � 3)

c. f(x) � (x � 3)3


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d. Describe the intersections of these graphs with the x-axis.

3. Using a graphing calculator, graph each quartic function. Then sketch its graph

on the grid.

a. f(x) � x(x � 2)2 (x � 2)

b. f(x) � �(x � 2)2 (x � 3)2

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c. f(x) � (x � 3)3 (x � 1)

d. Describe the intersections of these graphs with the x-axis.

e. How does the multiplicity of zeros affect the graph of a polynomial function?

Problem 3 ExtremaIn an earlier activity, you used the vertex of a quadratic function to determine the

lowest value of the function, the absolute minimum, or the highest value of the

function, the absolute maximum.

1. Examine all of the third-degree functions that you graphed in Problems 1 and 2.

Which have an absolute minimum or maximum? How do you know?

a. Is it possible for a third-degree polynomial function to have an absolute

minimum or maximum? Explain.


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b. Do any of the third-degree polynomial functions have points that are “similar”

to a maximum or a minimum? Explain.

2. Examine all of the fourth-degree functions that you graphed in Problems 1 and 2.

Which have an absolute minimum or maximum? How do you know?

a. Is it possible for a third-degree polynomial function to have an absolute

minimum or maximum? Explain.

b. Do any of the third-degree polynomial functions have points that are “similar”

to a vertex? Explain.

3. Points that are “similar” to vertices are called relative maximums or relative

minimums. Explain why.

4. Relative minimums and maximums are called extremum (singular) or extrema

(plural). Based on all of the polynomial functions you have graphed in this lesson,

how many extrema can a polynomial function have?

a. A quadratic function?

b. A cubic function?

c. A quartic function?

d. A polynomial function of degree n?

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5. Unfortunately, calculating extrema of most polynomial functions is difficult using

algebraic methods or the Calc function and is one of the major topics in

Calculus; however, using the zoom and trace functions on a graphing calculator,

we are able to approximate these points. For each of the following polynomial

functions, determine the zeros and extrema using your graphing calculator.

a. f(x) � x(x � 2)(x � 2)

Zeros: ________ Extrema: _____________________________

b. f(x) � �(x � 2)2 (x � 3)2

Zeros: _____ Extrema: _____________________________

c. f(x) � x4 � 3x3 � 9x2 � 3x � 10

Zeros: ______ Extrema: __________________________________________

Problem 4 End BehaviorPolynomial functions are continuous and well-behaved because there are no breaks

or sharp turns in their graphs—they are fairly predictable.

1. Examine the graphs of all the odd-degree polynomial functions in this lesson.

a. Describe what happens at either end of the graph.

b. How does changing the sign of the leading coefficient change this “end

behavior”?

2. Examine the graphs of all the even-degree polynomial functions.

a. Describe what happens at either end of the graph.

b. How does changing the sign of the leading coefficient change this “end

behavior”?


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3. Will this “end behavior” change if any of the other coefficients are much larger or

smaller than the leading coefficient? Explain.

4. Without graphing the following polynomial functions, describe their graphs,

including the number of zeros, the number of extrema, and their end behavior.

a. f(x) � �x2(x � 2)2 (x � 3)2

b. f(x) � 2x(x � 2)(x � 3)2 (x � 10)


Documents

5 Polynomial Functions...2011/08/30 · Approximations and Graphing p. 253 5.4 It’s Fundamental The Fundamental Theorem of Algebra p. 259 5.5 When Division Is Synthetic Polynomial