1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 2 Polynomial, Power, and Rational Functions

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.6 Complex Zeros and the Fundamental Theorem of Algebra

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

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

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 6 Fundamental Theorem of Algebra A polynomial function of degree n has n complex zeros (real and nonreal). Some of these zeros may be repeated.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 7 Linear Factorization Theorem

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

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 9 Example Exploring Fundamental Polynomial Connections

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 10 Example Exploring Fundamental Polynomial Connections

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

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 12 Complex Conjugate Zeros

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 13 Example Finding a Polynomial from Given Zeros

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Finding Complex Zeros Slide 2- 14

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 15

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 16 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.

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 17 Example Factoring a Polynomial

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 18 Example Factoring a Polynomial

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 19

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 20 HW: p234, #1-41 odd

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.7 Graphs of Rational Functions