1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.4 Real Zeros of Polynomial Functions

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 2 Quick Review

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 3 Quick Review Solutions

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 4 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.

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 5 Division Algorithm for Polynomials

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 6 Division Algorithm for Polynomials

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 7 Example Using Polynomial Long Division

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 8 Remainder Theorem

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 9 Example Using the Remainder Theorem

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 10 Factor Theorem

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 11 Example Using Synthetic Division

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 12 Descartes’ Rule of Signs Let P(x) be a polynomial over the real numbers. The number of positive roots of P(x) = 0 either equals the number of sign variations in P(x) or else is fewer by an even number. The number of negative roots of P(x) = 0 either equals the number of sign variations in P(–x) or else is fewer by an even number.

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 13 Rational Zeros Theorem

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 14 Upper and Lower Bound Tests for Real Zeros

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 15 Example Finding the Real Zeros of a Polynomial Function

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 16 Example Finding the Real Zeros of a Polynomial Function

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 17 Homework Homework #1 Read Section 2.5 Page 223, Exercises: 1 – 73 (EOO), skip 69

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.5 Complex Zeros and the Fundamental Theorem of Algebra

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 19 What you’ll learn about Two Major Theorems Complex Conjugate Zeros Factoring with Real Number Coefficients … and why These topics provide the complete story about the zeros and factors of polynomials with real number coefficients.

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 20 Fundamental Theorem of Algebra A polynomial function of degree n has n complex zeros (real and imaginary). Some of these zeros may be repeated.

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 21 Linear Factorization Theorem

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 22 Fundamental Polynomial Connections in the Complex Case The following statements about a polynomial function f are equivalent, if k is a complex number: 1. x = k is a solution (or root) of the equation f(x) = 0 2. k is a zero of the function f. 3. x – k is a factor of f(x).

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 23 Example Exploring Fundamental Polynomial Connections

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 24 Complex Conjugate Zeros

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 25 Example Finding a Polynomial from Given Zeros

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 26 Factors of a Polynomial with Real Coefficients Every polynomial function with real coefficients can be written as a product of linear factors and irreducible quadratic factors, each with real coefficients.

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 27 Example Factoring a Polynomial

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 28 Polynomial Functions of Odd Degree