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Chapter 6
Polynomials and
Polynomial Functions
In this chapter, you will …
Learn to write and graph polynomial
functions and to solve polynomial
equations.
Learn to use important theorems about
the number of solutions to polynomial
equations.
Learn to solve problems involving
permutations, combinations and
binomial probability.
6-1 Polynomial Functions
What you’ll learn …
To classify polynomials
To model data with polynomial functions
2.04 Create and use best-fit mathematical models of linear,
exponential, and quadratic functions to solve problems involving
sets of data.
a. Interpret the constants, coefficients, and bases in the context of the
data.
b. Check the model for goodness-of-fit and use the model, where
appropriate, to draw conclusions or make predictions.
2.06 Use cubic equations to model and solve problems.
a. Solve using tables and graphs.
b. Interpret constants and coefficients in the context of the problem.
A monomial is an expression that is a real number,
a variable or a product of real numbers and
variables.
13, 3x, -57, x², 4y², -2xy, or 520x²y²
A binomial is the sum of two monomials. It has
two unlike terms.
3x + 1, x² - 4x, 2x + y, or y - y²
A trinomial is the sum of three monomials. It has
three unlike terms.
x2 + 2x + 1, 3x² - 4x + 10,
2x + 3y + 2
A polynomial is the sum of one or more
terms. x2 + 2x, 3x3 + x² + 5x + 6, 4x - 6y + 8
• The exponent of the variable in a term
determines the degree of that term.
• The terms in the polynomial are in
descending order by degree.
• This order demonstrates the standard
form of a polynomial.
3
Leading
Coefficient
Cubic
Term
2
Quadratic
Term
Linear
Term
Constant
Term
Degree
Name Using
Degree
Polynomial
Example
Number
of
Terms
Name Using
Number of
Terms
0
Constant
6
1
monomial
1
Linear
x+3
2
binomial
2
Quadratic
3x2
1
monomial
3
Cubic
2x3 -5x2 -2x
3
trinomial
4
Quartic
x4 + 3x2
2
binomial
5
Quintic
-2x5+3x2-x+4
4
polynomial
Example 1 Classifying Polynomials
Write each in standard form and classify it by
degree and number of terms.
-7x + 5x4
x2 – 4x + 3x3 +2x
4x – 6x + 5
Linear Model
Quadratic Model
Cubic Model
Example 2 Comparing Models
Using a graphing calculator, determine whether
a linear model, a quadratic model, or a cubic
model best fits the values in the table.
x
0
5
10
15
20
y
10.1
2.8
8.1
16.0
17.8
Example p.303 #17
The data at the right
indicate that the life
expectancy for
residents of the US
has been increasing.
a. Find a quadratic model
for the data set.
b. Find a cubic model for
the data set.
c. Graph each model.
Compare the quadratic
and cubic models to
determine which one
is a better fit.
Year of
Birth
Males
Females
1970
67.1
74.7
1980
70.0
77.4
1990
71.8
78.8
2000
73.2
80.2
2010
74.5
81.3
Example p.303 #23
Find a cubic model for each function. Then
use your model to estimate the value of y
when x=17.
X
0
2
4
6
8
10
12
14
16
18
20
Y
4.1
6
15.7
21.1
23.6
23.1
24.7
24.9
23.9
25.2
29.5
Example p.304 #31
h=10
The diagram shows a
cologne bottle that
consists of a cylindrical
base and a hemispherical
top.
a. Write an expression for
the cylinder’s volume.
b. Write an expression for
the volume of the
hemispherical top.
r
c. Write a polynomial to
represent the total volume
6-2 Polynomials and Linear Factors
What you’ll learn …
To analyze the factored form of a polynomial
To write a polynomial function from its zeros
1.03 Operate with algebraic expressions
(polynomial, rational, complex fractions) to
solve problems.
Example 1 Writing a Polynomial in
Standard Form
Write the expression (x+1)(x+2)(x+3) as a
polynomial in standard form.
Write the expression (x+1)(x+1)(x+2) as a
polynomial in standard form.
Example 2 Writing a Polynomial in
Factored Form
Write 2x3 +10x2 + 12x in factored form.
Write 3x3 - 3x2 - 36x in factored form.
Example 3 Real World Connection
Several popular models of
a.
carry-on luggage have a
length 10 in. greater than
their depth. To comply
with airline regulations,
b.
the sum of the length,
width and depth may not
c.
exceed 40 in.
Assume that the sum of the
length, width and depth is 40 in.
Graph the function relating volume
V to depth x.
Describe a realistic domain.
What is the maximum possible
volume of a piece of luggage?
What are the corresponding
dimensions of the luggage?
The maximum value in Example 3 is the
greatest value of the points in a region of the
graph.
It is called a relative maximum.
Similarly, a relative minimum is the least
y-value among nearby points on a graph.
Relative maximum
Value of y
x intercepts
Relative minimum
Value of y
Example 4 Finding Zeros
of a Polynomial Function
Find the zeros of
y= (x-2)(x+1)(x+3).
Then graph the function.
Find the zeros of
y= (x-7)(x-5)(x-3).
Then graph the function.
You can reverse the process and write
linear factors when you know the zeros.
The relationship between the linear
factors of a polynomial and the zeros of a
polynomial is described by the Factor
Theorem.
Factor Theorem
The expression x-a is a linear factor of a
polynomial if and only if the value a is a zero
of the related polynomial function.
Example 5 Writing a Polynomial Function
From Its Zeros
Write a polynomial function in standard form with
zeros at -2, 3, and 3.
Write a polynomial function in standard form with
zeros at -4, -2, and 1.
While the polynomial function in
Example 5 has three zeros, it has only
two distinct zeros: -2 and 3. If a linear
factor of a polynomial is repeated, then
the zero is repeated. A repeated zero
is called a multiple zero. A multiple
zero has a multiplicity equal to the
number of times the zero occurs. In
example 5 the zero 3 has a multiplicity
of 2.
Example 6 Finding the Multiplicity
of a Zero
Find any multiple zeros of f(x)=x4 +6x3+8x2 and
state the multiplicity.
Find any multiple zeros of f(x)=x3 - 4x2+4x and state
the multiplicity.
Equivalent Statements
about Polynomials
1. -4 is a solution of x2 +3x -4 =0.
2. -4 is an x-intercept of the graph of
y= x2 +3x -4.
3. -4 is a zero of y= x2 +3x -4.
4. x+4 is a factor of x2 +3x -4.
6-3 Dividing Polynomials
What you’ll learn …
To divide polynomials using long division
To divide polynomials using synthetic division
1.03 Operate with algebraic expressions
(polynomial, rational, complex fractions) to
solve problems.
You can use polynomial division to help
find all the zeros of a polynomial
function. Division of polynomials is
similar to numerical long division.
8 56
217 65465
x
2
2x
Example 1a Polynomial Long Division
Divide x2 +3x – 12 by x - 3
Example 1b Polynomial Long Division
Divide x2 -3x + 1 by x - 4
Example 2 Checking Factors
Determine whether x+4 is a factor of each polynomial
x2 + 6x + 8
x3 + 3x2 -6x - 7
To check (divisor) (quotient) + r = dividend
To divide by a linear factor, you can
use a simplified process that is known
as synthetic division. In synthetic
division, you omit all variables and
exponents. By reversing the sign in
the divisor, you can add throughout the
process instead of subtracting.
Example 3a Using Synthetic Division
Use synthetic division to divide
3x3 – 4x2 +2x – 1 by x +1
Example 3b Using Synthetic Division
Use synthetic division to divide
x3 + 4x2 + x – 6 by x +1
Example 4 Real World Connection
The volume in cubic feet
of a sarcophagus
(excluding the cover) can
be expressed as the
product of its three
dimensions:
V(x) = x3 – 13x + 12. The
length is x + 4.
a. Find linear expressions
with integer coefficients
for the other dimensions.
Assume that the width is
greater than the height.
Check for Understanding
Use synthetic division to divide
x3 - 2x2 - 5x + 6 by x + 2
Remainder Theorem
If a polynomial P(x) of degree n>1 is
divided by (x-a). Where a is a constant,
then the remainder is P(a).
Example 5a Evaluating a Polynomial by
Synthetic Division
Use synthetic division to find P(-4)
for P(x) = x4 - 5x2 + 4x + 12
Example 5b Evaluating a Polynomial by
Synthetic Division
Use synthetic division to find P(-1)
for P(x) = 2x4 + 6x3 – x2 - 60
6-4 Solving Polynomial Equations
What you’ll learn …
To solve polynomials equations by graphing
To solve polynomials equations by factoring
1.03 Operate with algebraic expressions
(polynomial, rational, complex fractions) to
solve problems.
Example 1 Solving by Graphing
Steps
 Graph y1= x3 + 3x2
 Graph y2= x + 3
 Use the intersection
feature to find the x
Solve x3 - 19x = -2x2 + 20
values at the points
of intersection.
Solve x3 + 3x2 = x + 3
Example 3 Real World Connection
The dimensions in inches of a portable
kennel can be expressed as width x,
length x+7 and height x-1. The
volume is 5.9 ft3. Find the portable
kennel’s dimensions.
Sometimes you can solve polynomial
equations by factoring the polynomial
and using the factor Theorem. Recall
that a quadratic expression that is the
difference of squares has a special
factoring pattern. Similarly, a cubic
expression may be the sum of cubes or
the difference of cubes.
Sum and Differences of Cubes
a3 + b3 = (a + b)(a2 – ab + b2)
a3 - b3 = (a - b)(a2 + ab + b2)
Example 3 Factoring a Sum of Cubes
Factor 8x3 + 1
Factor 64x3 + 27
Example 3 Factoring a Difference of Cubes
Factor 8x3 - 27
Factor 125x3 - 64
Example 4 Solving a Polynomial Equation
Factor x3 + 8 = 0
Factor 27x3 + 1
Example 5 Factoring by Using a Quadratic Form
Factor x4 - 2x2 – 8 = 0
Factor x4 + 7x2 + 6 = 0
6-5 Theorems About Roots of
Polynomial Equations
What you’ll learn …
To solve equations using the Rational Root Theorem
To use the Irrational Root Theorem and the
Imaginary Root Theorem
1.02 Define and compute with complex
numbers.
1.03 Operate with algebraic expressions
(polynomial, rational, complex fractions) to
solve problems.
Consider the equivalent
equations….
x3 – 5x2 -2x +24 = 0 and (x+2)(x-3)(x-4) =0
-2, 3 and 4 are the roots of the equation.
• The product of -2,3 and 4 is 24.
• Notice that all the roots are factors of the
constant term 24.
• In general, if the coefficients in a polynomial
equation are integers, then any integer root of
the equation is a factor of the constant term.
Both the constant and the leading coefficient of a
polynomial can play a key role in identifying the rational
roots of the related polynomial equation. The role is
expressed in the Rational Root Theorem.
p
If q is in simplest form and is a
rational root of the polynomial equation
with integer coefficients, then p must
be a factor of the constant term and q
must be a factor of the leading
coefficient.
Example 1a Finding Rational Roots
x3
-
4x2
- 2x + 8 = 0
1.
2.
Steps
List the possible rational
roots of the leading
coefficient and the
constant.
Test each possible root.
Example 1b Finding Rational Roots
2x3
-
x2
+2x - 1 = 0
1.
2.
Steps
List the possible rational
roots of the leading
coefficient and the
constant.
Test each possible root.
Example 1c Finding Rational Roots
x3
-
2x2
- 5x + 10 = 0
1.
2.
Steps
List the possible rational
roots of the leading
coefficient and the
constant.
Test each possible root.
Example 1d Finding Rational Roots
3x3
+
x2
-x+1=0
1.
2.
Steps
List the possible rational
roots of the leading
coefficient and the
constant.
Test each possible root.
In Chapter 5 you learned to find irrational
solutions to quadratic equations. For
example, by the Quadratic Formula,
the solutions of x2 – 4x -1 =0 are
2+√5 and 2 - √5.
Number pairs of the form a+√b and a-√b
are called conjugates.
You can often use conjugates to find the
irrational roots of a polynomial equation.
Irrational Root Theorem
Let a and b be rational numbers and
let √b be an irrational number. If
a+ √b is a root of a polynomial
equation with rational coefficients, then
the conjugate a- √b also is a root.
Example 3 Finding Irrational Roots
A polynomial equation with integer coefficients has
the following roots. Find two additional roots..
1 + √3 and -√11
2 - √7 and √5
Number pairs of the form a+bi and a-bi are
complex conjugates. You can use complex
conjugates to find an equation’s imaginary roots.
Imaginary Root Theorem
If the imaginary number a+bi is a root
of a polynomial equation with real
coefficients then the conjugate a-bi
also is a root.
Example 4 Finding Imaginary Roots
A polynomial equation with integer coefficients has
the following roots. Find two additional roots..
3i and -2 + i
3 - i and 2i
Example 5a Writing a Polynomial
Equation from its Roots
Find a third degree polynomial equation with
rational coefficients that has roots -1 and 2-i.
1.
2.
3.
Steps
Find the other root using
the Imaginary Root
Theorem.
Write the factored form
of the polynomial using
the Factor Theorem.
Multiply the factors.
Example 5b Writing a Polynomial
Equation from its Roots
Find a third degree polynomial equation with
rational coefficients that has roots 3 and 1+i.
1.
2.
3.
Steps
Find the other root using
the Imaginary Root
Theorem.
Write the factored form
of the polynomial using
the Factor Theorem.
Multiply the factors.
Example 5c Writing a Polynomial
Equation from its Roots
Find a fourth degree polynomial equation with
rational coefficients that has roots i and 2i.
1.
2.
3.
Steps
Find the other root using
the Imaginary Root
Theorem.
Write the factored form
of the polynomial using
the Factor Theorem.
Multiply the factors.
6-6 The Fundamental Theorem
of Algebra
What you’ll learn …
To use the Fundamental Theorem of Algebra in
solving polynomial equations with complex roots
1.02 Define and compute with complex
numbers.
1.03 Operate with algebraic expressions
(polynomial, rational, complex fractions) to
solve problems.
You have solved polynomial equations
and found that their roots are included
in the set of complex numbers. That
is, the roots have been integers,
rational numbers, irrational numbers
and imaginary numbers.
But can all polynomial equations be
solved using complex numbers?
In 1799, the German
mathematician Carl Friedrich
Gauss proved that the answer
to this question is yes. The
roots of every polynomial
equation, even those with
imaginary coefficients, are
complex numbers.
Carl Friedrich Gauss
The answer is so important
that his theorem is called the
Fundamental Theorem of
Algebra.
Fundamental Theorem of Algebra
If P(x) is a polynomial of degree n>1
with complex coefficients, then P(x) = 0
has at least one complex root.
Corollary
Including imaginary roots and multiple
roots, an nth degree polynomial
equation has exactly n roots; the
related polynomial function has exactly
n zeros.
Example 1a Using the Fundamental
Theorem of Algebra
Find the number of complex roots, the
possible number of real roots and possible
number of rational roots.
x4 - 3x3 + x2 – x +3 = 0
Example 1b Using the Fundamental
Theorem of Algebra
Find the number of complex roots, the
possible number of real roots and possible
number of rational roots.
x3 - 2x2 + 4x -8 = 0
Example 1c Using the Fundamental
Theorem of Algebra
Find the number of complex roots, the
possible number of real roots and possible
number of rational roots.
x5 + 3x4 - x - 3 = 0
6-8 The Binomial Theorem
What you’ll learn …
To use Pascal’s Triangle
To use the Binomial Theorem

1.03 Operate with algebraic expressions
(polynomial, rational, complex fractions)
to solve problems.
You have learned to multiply binomials
using the FOIL method and the
Distributive Property. If you are raising
a single binomial to a power, you have
another option for finding the product.
Consider the expansion of several
binomials. To expand a binomial
being raised to a power, first multiply;
then write the result as a polynomial in
standard form.
Pascal’s Triangle
(a + b)2
=
(a + b) (a + b)
a2 + 2ab + b2
The coefficients of the product are 1,2 1.
(a + b)3
=
(a + b) (a + b) (a + b)
a3 + 3a2b + 3ab2 + b3
The coefficients of the product are 1,3,3,1.
Pascal’s Triangle
Example 1a Using Pascal’s Triangle
Expand (a+b)8
Example 1b Using Pascal’s Triangle
Expand (x - 2)4
Example 1c Using Pascal’s Triangle
Expand (m + 3)5
Example 1d Using Pascal’s Triangle
Expand (3 – 2x)6
Quadratic
Inequalities
Quadratics
Before we get started let’s review.
A quadratic equation is an equation that can
2
be written in the form ax  bx  c  0 ,
where a, b and c are real numbers and a cannot equal
zero.
In this lesson we are going to discuss quadratic
inequalities.
Quadratic Inequalities
What do they look like?
Here are some examples:
x 2  3x  7  0
3x  4 x  4  0
2
x  16
2
Quadratic Inequalities
When solving inequalities we are trying to
find all possible values of the variable
which will make the inequality true.
Consider the inequality
x2  x  6  0
We are trying to find all the values of x for which the
quadratic is greater than zero or positive.
Solving a quadratic inequality
We can find the values where the quadratic equals zero
2
x
 x6  0
by solving the equation,
x  3x  2  0
x  3  0 or x  2  0
x  3 or x  2
Solving a quadratic inequality
2
x
 x6  0
For the quadratic inequality,
we found zeros 3 and –2 by solving the equation
x 2  x  6  0. Put these values on a number line and
we can see three intervals that we will test in the
inequality. We will test one value from each interval.
-2
3
Solving a quadratic inequality
Interval
 ,2
Test Point
Evaluate in the inequality
True/False
x2  x  6  0
x  3
 32   3  6  9  3  6  6  0 True
x2  x  6  0
 2, 3
 3, 
x0
02  0  6  0  0  6  6  0
False
x2  x  6  0
x4
42  4  6  16  4  6  6  0
True
Example 2:
2
2
x
 3x  1  0
Solve
2
2
x
 3x  1  0
First find the zeros by solving the equation,
2 x 2  3x  1  0
2x 1x 1  0
2 x  1  0 or x  1  0
1
x  or x  1
2
Example 2:
Now consider the intervals around the zeros and test
a value from each interval in the inequality.
The intervals can be seen by putting the zeros on a
number line.
1/2
1
Example 2:
Interval
Test Point
Evaluate in Inequality
True/False
2 x 2  3x  1  0
1

  , 
2

x0
20  30  1  0  0  1  1  0
2
False
2 x 2  3x  1  0
1 
 ,1
2 
9 9
1
3 3
2   3   1    1   0
8 4
8
4 4
2
3
x
4
True
2 x 2  3x  1  0
1, 
x2
22  32  1  8  6  1  3  0
2
False
Summary
In general, when solving quadratic inequalities
1. Find the zeros by solving the equation you get
when you replace the inequality symbol with an
equals.
2. Find the intervals around the zeros using a number
line and test a value from each interval in the
number line.
3. The solution is the interval or intervals which make
the inequality true.
x  5 x  24  0
2
12  x  x  0
2
16 x  1  0
2
x  5x  4  0
2
In this chapter, you should have …
Learned to write and graph polynomial
functions and to solve polynomial
equations.
Learned to use important theorems
about the number of solutions to
polynomial equations.
Learned to solve problems involving
permutations, combinations and
binomial probability.