Section 5.5

The Real Zeros of a Polynomial Function

Use the Remainder and Factor Theorem

Division Algorithm for Polynomials

Let f be a polynomial function. If we divide f (x)

by x c, then the remainder is f (c).

Remainder Theorem

3 2Find the remainder if 3 2 1 isdivided by

(a) 2 (b) 1

f x x x x

x x

3 2(a) 2 2 3 2 2 2 1 1f

The remainder is 1.

3 21(b) 1 1 1 3 2 1 5f

The remainder is 5.

Using the Remainder Theorem

The Factor Theorem

That is,

3 2

Use the Factor Theorem to determine whether the function2 4 3 has the factor

(a) 1 (b) 1

f x x x x

x x

3 2(a) 1 2 1 1 4 1 3 0f

By the factor theorem, x + 1 is a factor of f (x).

3 2(b) 1 2 1 1 4 1 3 4f

By the factor theorem, x 1 is not a factor of f (x).

Using the Factor Theorem

Theorem

Using the Rational Zeros Theorem to List the Potential

Rational Zeros of a Polynomial Function

Theorem

3 2

List the potential rational zeros of3 8 7 12f x x x x

Factors of the constant

Factors of the leading coefficient

: 1, 2, 3, 4, 6, 12p : 1, 3q 1 2 4

: 1, 2, 3, 4, 6, 12, , ,3 3 3

p

q

Example

Find the Real Zeros of a Polynomial Function

1 2 4: 1, 2, 3, 4, 6, 12, , ,

3 3 3

p

q

3 2Find the real zeros of the polynomial function 3 8 7 12.Write in factored form.

f x x x xf

Since f is a polynomial of degree 3, there are at most three real zeros.

21 3 5 12f x x x x

1 3 3 4x x x 1 3 3 4 0x x x

41 or 3 or

3x x x

Summary of the Method

4 3 2Find the real zeros of 2 13 29 27 9.Write in factored form.

f x x x x xf

Step 1: There are at most 4 real zeros.1 3 9

The potential rational zeros are : 1, , 3, , 9,2 2 2

p

q Step 2:

3 2 1 2 11 18 9f x x x x x

2 2 1 2 9 9f x x x x

2 1 3 2 3f x x x x

31 or 3 or

2x x x 2

1 3 2 3 0x x x

Solving Polynomial Equations

4 3 2Solve the equation: 2 13 29 27 9 0x x x x

21 2 3 3 0x x x

31, , 3

2

Solving a Polynomial Equation

The solution set is given by

Two Theorems

Theorem on Bounds of Zeros

Bound

Example on Using the Theorem

Example on Using the Theorem

Example on Using the Theorem

The Intermediate Value Theorem

The Intermediate Value Theorem

Using the Intermediate Value Theorem to Locate a Zero

Approximating Zeros

Example: Approximating Zeros