Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 2 Polynomial, Power, and Rational Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.6 Complex Zeros and the Fundamental Theorem of Algebra
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 4 Quick Review Solutions
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.
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 7 Linear Factorization Theorem
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).
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 9 Example Exploring Fundamental Polynomial Connections
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Exploring Fundamental Polynomial Connections
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 11
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Complex Conjugate Zeros
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding a Polynomial from Given Zeros
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Finding Complex Zeros Slide 2- 14
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Factoring a Polynomial
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Factoring a Polynomial
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 19
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide HW: p234, #1-41 odd
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.7 Graphs of Rational Functions