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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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Lesson 5.1 ● Introduction to Polynomial Expressions, Equations, and Functions 239
5
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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Lesson 5.1 ● Introduction to Polynomial Expressions, Equations, and Functions 241
5
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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Lesson 5.1 ● Introduction to Polynomial Expressions, Equations, and Functions 243
5
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
ObjectivesIn this lesson, you will:
● 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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Lesson 5.2 ● Solving Polynomial Equations and Inequalities: Factoring 247
5
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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Lesson 5.2 ● Solving Polynomial Equations and Inequalities: Factoring 249
5
b. 0 � (2x � 4)(x � 1)(5 � 3x)
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c. x4 � 4x3 � x2 � 4x � 0
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Lesson 5.2 ● Solving Polynomial Equations and Inequalities: Factoring 251
5
Be prepared to share your work with another pair, group, or the entire class.
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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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Lesson 5.3 ● Solving Polynomial Equations: Approximations and Graphing 255
5
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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Lesson 5.3 ● Solving Polynomial Equations: Approximations and Graphing 257
5
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
Be prepared to share your work with another pair, group, or the entire class.
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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
ObjectivesIn this lesson, you will
● 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.
Be prepared to share your work with another pair, group, or the entire class.
262 Chapter 5 ● Polynomial Functions
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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.
ObjectivesIn this lesson, you will
● 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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Lesson 5.5 ● Polynomial and Synthetic Division 265
5
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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Lesson 5.5 ● Polynomial and Synthetic Division 267
5
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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Lesson 5.5 ● Polynomial and Synthetic Division 269
5
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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Lesson 5.5 ● Polynomial and Synthetic Division 271
5
b. f(x) � 3x3 � 4x2 � 8 divided by x � 2
c.
d.
Calculate
Be prepared to share your work with another pair, group, or the entire class.
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
ObjectivesIn this lesson, you will
● 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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Lesson 5.6 ● The Remainder and Factor Theorems 275
5
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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Lesson 5.6 ● The Remainder and Factor Theorems 277
5
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.
Be prepared to share your work with another pair, group, or the entire class.
x � 1� i, �1
2,
3
2
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Lesson 5.7 ● Extrema and End Behavior 279
5
ObjectivesIn this lesson, you will:
● 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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Lesson 5.7 ● Extrema and End Behavior 281
5
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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Lesson 5.7 ● Extrema and End Behavior 283
5
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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Lesson 5.7 ● Extrema and End Behavior 285
5
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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Lesson 5.7 ● Extrema and End Behavior 287
5
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)
Be prepared to share your work with another pair, group, or the entire class.