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

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Presentation transcript:

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

Copyright © 2011 Pearson, Inc. Slide 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 © 2011 Pearson, Inc. Slide 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 © 2011 Pearson, Inc. Slide Linear Factorization Theorem

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

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

Copyright © 2011 Pearson, Inc. Slide Complex Conjugate Zeros

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

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

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

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

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

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

Copyright © 2011 Pearson, Inc. Slide Quick Review

Copyright © 2011 Pearson, Inc. Slide Quick Review Solutions