1. Copyright © 2011 Pearson, Inc. 2.4 Real Zeros of Polynomial Functions

2. Copyright © 2011 Pearson, Inc. Slide 2.4 - 2 What you’ll learn about Long Division and the Division Algorithm Remainder and Factor Theorems Synthetic Division Rational Zeros Theorem Upper and Lower Bounds … and why These topics help identify and locate the real zeros of polynomial functions.

3. Copyright © 2011 Pearson, Inc. Slide 2.4 - 3 Division Algorithm for Polynomials

4. Copyright © 2011 Pearson, Inc. Slide 2.4 - 4 Example Using Polynomial Long Division

5. Copyright © 2011 Pearson, Inc. Slide 2.4 - 5 Example Using Polynomial Long Division

6. Copyright © 2011 Pearson, Inc. Slide 2.4 - 6 Remainder Theorem

7. Copyright © 2011 Pearson, Inc. Slide 2.4 - 7 Example Using the Remainder Theorem

8. Copyright © 2011 Pearson, Inc. Slide 2.4 - 8 Example Using the Remainder Theorem

9. Copyright © 2011 Pearson, Inc. Slide 2.4 - 9 Factor Theorem

10. Copyright © 2011 Pearson, Inc. Slide 2.4 - 10 Fundamental Connections for Polynomial Functions For a polynomial function f and a real number k the following statements are equivalent: 1. x = k is a solution (or root) of the equation f(x) = 0 2. k is a zero of the function f. 3. k is an x-intercept of the graph of y = f(x). 4. x – k is a factor of f(x).

11. Copyright © 2011 Pearson, Inc. Slide 2.4 - 11 Example Using Synthetic Division

12. Copyright © 2011 Pearson, Inc. Slide 2.4 - 12 Example Using Synthetic Division

13. Copyright © 2011 Pearson, Inc. Slide 2.4 - 13 Rational Zeros Theorem

14. Copyright © 2011 Pearson, Inc. Slide 2.4 - 14 Upper and Lower Bound Tests for Real Zeros

15. Copyright © 2011 Pearson, Inc. Slide 2.4 - 15 Example Finding the Real Zeros of a Polynomial Function

16. Copyright © 2011 Pearson, Inc. Slide 2.4 - 16 Example Finding the Real Zeros of a Polynomial Function

17. Copyright © 2011 Pearson, Inc. Slide 2.4 - 17 Example Finding the Real Zeros of a Polynomial Function

18. Copyright © 2011 Pearson, Inc. Slide 2.4 - 18 Example Finding the Real Zeros of a Polynomial Function

19. Copyright © 2011 Pearson, Inc. Slide 2.4 - 19 Example Finding the Real Zeros of a Polynomial Function

20. Copyright © 2011 Pearson, Inc. Slide 2.4 - 20 Example Finding the Real Zeros of a Polynomial Function

21. Copyright © 2011 Pearson, Inc. Slide 2.4 - 21 Example Finding the Real Zeros of a Polynomial Function

22. Copyright © 2011 Pearson, Inc. Slide 2.4 - 22 Example Finding the Real Zeros of a Polynomial Function

23. Copyright © 2011 Pearson, Inc. Slide 2.4 - 23 Example Finding the Real Zeros of a Polynomial Function

24. Copyright © 2011 Pearson, Inc. Slide 2.4 - 24 Quick Review

25. Copyright © 2011 Pearson, Inc. Slide 2.4 - 25 Quick Review Solutions