1 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Start Up Day 13.

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Presentation transcript:

1 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Start Up Day 13

2 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 2.5 Complex Zeros and the Fundamental Theorem of Algebra Demana, Waits, Foley, Kennedy

3 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. OBJECTIVE: Students will be able to determine apply the fundamental theorem of Algebra & the Complex Conjugate Theorem to determine all roots to a polynomial function. ESSENTIAL QUESTION: How do these two theorems help to complete our understanding about the zeros and factors of polynomials with real number coefficients? CLASS-WORK/HOME LEARNING: pg. 215 #2, 11, 16, 22, 29, 38/ pg. 215 #4, 7, 23, 31, 33, 39 + MATHXL Module B Quiz #1 Review (Lessons 2.1 through 2.5)

4 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Fundamental Theorem of Algebra A polynomial function of degree n has n complex zeros (real and nonreal). Some of these zeros may be repeated.

5 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Linear Factorization Theorem

6 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Let ’ s review… think ‘ p / q ’ Find all factors of the given function: https :// youtu. be / pYuiVXdhVSo

7 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 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).

8 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Exploring Fundamental Polynomial Connections

9 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

10 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Visualizing Complex Solutions within the Imaginary Plane!

11 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Complex Conjugate Zeros

12 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Finding a Polynomial from Given Zeros

13 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

14 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution continued

15 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 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.

16 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Factoring a Polynomial

17 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

18 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution continued