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Copyright © 2011 Pearson, Inc. 2.5 Complex Zeros and the Fundamental Theorem of Algebra.

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Presentation on theme: "Copyright © 2011 Pearson, Inc. 2.5 Complex Zeros and the Fundamental Theorem of Algebra."— Presentation transcript:

1 Copyright © 2011 Pearson, Inc. 2.5 Complex Zeros and the Fundamental Theorem of Algebra

2 Copyright © 2011 Pearson, Inc. Slide 2.5 - 2 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.

3 Copyright © 2011 Pearson, Inc. Slide 2.5 - 3 Fundamental Theorem of Algebra A polynomial function of degree n has n complex zeros (real and nonreal). Some of these zeros may be repeated.

4 Copyright © 2011 Pearson, Inc. Slide 2.5 - 4 Linear Factorization Theorem

5 Copyright © 2011 Pearson, Inc. Slide 2.5 - 5 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).

6 Copyright © 2011 Pearson, Inc. Slide 2.5 - 6 Example Exploring Fundamental Polynomial Connections

7 Copyright © 2011 Pearson, Inc. Slide 2.5 - 7 Example Exploring Fundamental Polynomial Connections

8 Copyright © 2011 Pearson, Inc. Slide 2.5 - 8 Complex Conjugate Zeros

9 Copyright © 2011 Pearson, Inc. Slide 2.5 - 9 Example Finding a Polynomial from Given Zeros

10 Copyright © 2011 Pearson, Inc. Slide 2.5 - 10 Example Finding a Polynomial from Given Zeros

11 Copyright © 2011 Pearson, Inc. Slide 2.5 - 11 Example Finding a Polynomial from Given Zeros

12 Copyright © 2011 Pearson, Inc. Slide 2.5 - 12 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.

13 Copyright © 2011 Pearson, Inc. Slide 2.5 - 13 Example Factoring a Polynomial

14 Copyright © 2011 Pearson, Inc. Slide 2.5 - 14 Example Factoring a Polynomial

15 Copyright © 2011 Pearson, Inc. Slide 2.5 - 15 Example Factoring a Polynomial

16 Copyright © 2011 Pearson, Inc. Slide 2.5 - 16 Quick Review

17 Copyright © 2011 Pearson, Inc. Slide 2.5 - 17 Quick Review Solutions

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