**PURPOSE**

This lesson covers polynomial functions. The graphing, writing, and use of the
functions are

stressed.

**OBJECTIVES**

After completing this lesson, you should be able to

• identify a polynomial function;

• evaluate a polynomial function using synthetic division and determine
its zeros;

• use synthetic division to apply the remainder and factor theorems;

• graph a polynomial function and determine an equation for a polynomial
graph;

• write a polynomial function for a given situation and find the maximum
or minimum value of

the function;

• use technology to approximate the real roots of a polynomial equation;

• solve polynomial equations by various methods of factoring and the
RATIONAL ROOT

THEOREM; and

• apply several theorems about polynomial functions.

**READING ASSIGNMENT
**

Chapter 2, Sections 2-1 through 2-7 (pages 53–93)

Section 2-1: Zeros and Factors of Polynomial Functions, pp. 53–58

A polynomial function involves an equation that can be written in the following form:

The powers of x can be any number, and they should be
written in decreasing order. Examples of

polynomials follow with key terms for each.

Equation |
Degree |
coefficient |
Constant |
Leading coefficient |
Terms |

The roots of a polynomial function are x-values that give
the function a value of zero. The roots of

P (x) = 3 x − 7 and P (x) = 3 x^{2} + 2 x − 8 are found below:

is a root and a zero
for P (x) = 3 x − 7.

and −2 are roots and zeros for P (x) = 3 x^{2}
+ 2 x − 8.

Other values of the function can be found by placing desired values into the
function.

**Example 1:** Find P (−3) given P (x) = 2 x^{2} − 4 x + 6:

**Example 2:** Let's try problem 18b on page 56 of your text.

Find k (1 + i) given k(x) = x^{2} (x^{2} + 16)

Synthetic substitution is another method that can be used
to find values of functions. I will use

synthetic substitution to determine the value of f (−5) when f (x) = 2 x^{3} − 3 x^{2}
+ 4 x + 5.

List all of the coefficients. Drop the leading coefficient down, multiply 2 by
−5, and place in the

box:

Add −3 and −10. Multiply −13 by −5 and place in the box:

Add 4 and 65. Multiply 69 by −5 and place in the box:

Add 5 and −345:

The result is −340, so f (−5) = − 340.

Let's check it out:

For this problem, synthetic substitution would have been
the quicker method. The synthetic

substitution can be done in one step as follows

**Example 3:** Use synthetic substitution to find P (−2) when
P (x) = x^{5} − 1.

There are several missing terms in x^{5} − 1. The complete polynomial equation is

x^{5} + 0 x^{4} + 0 x^{3} + 0 x^{2} + 0 x − 1. All terms must be used in synthetic
substitution:

In this problem P (−2) = (−2)^{5} − 1, but −32 − 1 = − 33 is
quicker. Which method would be

quicker for P (−1) = x^{15} − 2? Think of all the missing terms! This would be
quicker using

substitution. Use the method best suited for the situation.

**Example 4: **Let's try problem 24 on page 57 of your text.

If 2i is a zero, then f (2i) = 0:

**Example 5: **Let's try problem 30b on page 57 of your text.

Given g (x) = 3 − 8 x, find g (x + 2) − g (x):

**Study Exercises
**

Complete odd-numbered problems 1–27 in the Written Exercises section on pages 56–57 of your

text. Then check your answers in the back of the text.

Synthetic substitution is versatile. Used not only to find the value of a function, it can also be used to

find the quotient and remainder in a polynomial division problem. When it is used in this manner, it

is referred to as synthetic division. I will work through a polynomial division problem and compare

the results to those in a corresponding synthetic division.

Multiply | |||

subtract | |||

subtract | Multiply | ||

Multiply | |||

subtract | |||

remind |

Since I divided by x − 2, I will use 2 in my synthetic
division. If I had divided by x + 3, I would

have used −3 in my synthetic division:

The remainder is 4 and 3 x^{2} + 1 x + 3 is the quotient.

If the binomial x − 2 would have been a factor of 3 x^{3} − 5 x^{2} + x − 2, the
remainder would have

been zero. So if my only desire is to determine if x − 2 is a factor of 3 x^{3} − 5
x^{2} + x − 2, I don't need

to divide it out. I could use synthetic division and just look at the remainder.
Or, I could use

substitution. In this case, I would find P(2) given P(x) = 3 x^{3} − 5 x^{2} + x −
2:

P(2) = 3 (2)^{3} − 5 (2)^{2} + 2 − 2.

P(2) = 4 is the same as the remainder.

This demonstrates the REMAINDER THEOREM.

REMAINDER THEOREMWhen a polynomial P (x) is divided by x − a, the remainder is P (a) |

So, depending on the problem, I could choose synthetic
division or substitution to determine if x − a

is a factor of P(x). If I wish to divide a polynomial by a polynomial of a
degree higher than 1, I need

to use long division.