1. Chapter 4: Polynomial and Rational Functions

2. Determine the roots of the polynomial 4-4 The Rational Root Theorem x 2 + 2x – 8 = 0

3. 4-4 The Rational Root Theorem Rational Root Theorem: List the possible rational roots of: 4x 3 – 2x 2 + x – 3

4. Example 1: List the possible roots of the equation. Then determine the rational roots. 3x 3 - 13x 2 + 2x + 8 = 0 4-4 The Rational Root Theorem

5. Problem 1: List the possible roots of the equation. Then determine the rational roots. x 3 + 7x 2 + 14x + 8 = 0 4-4 The Rational Root Theorem

6. Problem 2: List the possible roots of the equation. Then determine the rational roots. x 4 – x 3 + 2x 2 – 4x – 8 = 0 4-4 The Rational Root Theorem

7. Problem 4: List all the possible rational roots of the following polynomial equation, then find all the roots. x 3 + 6x 2 + 4x + 24 = 0 4-4 The Rational Root Theorem

8. f(x) = x 4 – x 3 + 2x 2 – 4x – 8 4-4 The Rational Root Theorem Descartes’ Rule of signs circa 1637 If P(x) is a polynomial in descending order, then the number of positive real zeros of P(x) is the same as the number of sign changes of the coefficients, or less by and even number. The number of negative real zeros of P(x) is the same as the number of sign changes of the coefficients of P(-x), or less by and even number..

9. 4-4 The Rational Root Theorem Example 2: Find the number of possible positive real zeros and the number of possible negative real zeros for the function and then find the real roots. f(x) = x 5 – x 4 – 8x 3 + 8x 2 – 9x + 9

10. HW 4-4 pg. 234 #11-21 odd and #25 4-4 The Rational Root Theorem

11. Problem 3: Find the number of possible positive real zeros and the number of possible negative real zeros for the function g(x) = 3x 4 + 16x 3 – 7x 2 – 64x – 20 4-4 The Rational Root Theorem