1. Section 5.5 The Real Zeros of a Polynomial Function

2. Use the Remainder and Factor Theorem

3. Division Algorithm for Polynomials

4. Let f be a polynomial function. If we divide f (x) by x  c, then the remainder is f (c). Remainder Theorem

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

6. The Factor Theorem That is,

7. 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

8. Theorem

9. Using the Rational Zeros Theorem to List the Potential Rational Zeros of a Polynomial Function

10. Theorem

11. Factors of the constant Factors of the leading coefficient Example

12. Find the Real Zeros of a Polynomial Function

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

14. Summary of the Method

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

16. Solving Polynomial Equations

17. Solving a Polynomial Equation The solution set is given by

18. Two Theorems

19. Theorem on Bounds of Zeros

20. Bound

21. Example on Using the Theorem

24. The Intermediate Value Theorem

26. Using the Intermediate Value Theorem to Locate a Zero

27. Approximating Zeros

28. Example: Approximating Zeros