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

The remainder is  1. The remainder is 5. Using the Remainder Theorem

The Factor Theorem That is,

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

Factors of the constant Factors of the leading coefficient Example

Find the Real Zeros of a Polynomial Function

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

Summary of the Method

Step 1: There are at most 4 real zeros. Step 2:

Solving Polynomial Equations

Solving a Polynomial Equation The solution set is given by

Two Theorems

Theorem on Bounds of Zeros

Bound

Example on Using the Theorem

The Intermediate Value Theorem

Using the Intermediate Value Theorem to Locate a Zero

Approximating Zeros

Example: Approximating Zeros