2 5 Complex Zeros and the Fundamental Theorem

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2. 5 Complex Zeros and the Fundamental Theorem of Algebra Copyright © 2011 Pearson,

2. 5 Complex Zeros and the Fundamental Theorem of Algebra Copyright © 2011 Pearson, Inc.

What you’ll learn about n n n Two Major Theorems Complex Conjugate Zeros Factoring

What you’ll learn about n n n 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. 2

Fundamental Theorem of Algebra A polynomial function of degree n has n complex zeros

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. 3

Linear Factorization Theorem Copyright © 2011 Pearson, Inc. 4

Linear Factorization Theorem Copyright © 2011 Pearson, Inc. 4

Fundamental Polynomial Connections in the Complex Case The following statements about a polynomial function

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. 5

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

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

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

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

Complex Conjugate Zeros Copyright © 2011 Pearson, Inc. 8

Complex Conjugate Zeros Copyright © 2011 Pearson, Inc. 8

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

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

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

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

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

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

Factors of a Polynomial with Real Coefficients Every polynomial function with real coefficients can

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. 12

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

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

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

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

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

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

Quick Review Copyright © 2011 Pearson, Inc. 16

Quick Review Copyright © 2011 Pearson, Inc. 16

Quick Review Solutions Copyright © 2011 Pearson, Inc. 17

Quick Review Solutions Copyright © 2011 Pearson, Inc. 17

Section 2 5 Complex Zeros and Fundamental TheoremSection 2 5 Complex Zeros and Fundamental Theorem
Section 5 6 Complex Zeros Fundamental Theorem ofSection 5 6 Complex Zeros Fundamental Theorem of
3 5 Complex Zeros the Fundamental Theorem of3 5 Complex Zeros the Fundamental Theorem of
THE FUNDAMENTAL THEOREM OF CALCULUS The Fundamental TheoremTHE FUNDAMENTAL THEOREM OF CALCULUS The Fundamental Theorem
3 Fundamental Theorem of Calculus Fundamental Theorem of3 Fundamental Theorem of Calculus Fundamental Theorem of
Finding Zeros of a Polynomial Function Fundamental TheoremFinding Zeros of a Polynomial Function Fundamental Theorem
Complex Numbers the Complex Plane Demoivres Theorem ComplexComplex Numbers the Complex Plane Demoivres Theorem Complex
3 Significant zeros Deciding whether zeros are significant3 Significant zeros Deciding whether zeros are significant
11 2 Zeros of Polynomial Functions Zeros of11 2 Zeros of Polynomial Functions Zeros of
Z COMPLEX R O S Complex zeros orZ COMPLEX R O S Complex zeros or
SHAPE SPACE Pythagorean Theorem Pythagorean Theorem Pythagorean TheoremSHAPE SPACE Pythagorean Theorem Pythagorean Theorem Pythagorean Theorem
LEMIs Theorem Lamis Theorem Statement Lamis theorem isLEMIs Theorem Lamis Theorem Statement Lamis theorem is
VARIGNONS THEOREM What is Varigons Theorem The theoremVARIGNONS THEOREM What is Varigons Theorem The theorem
Thvenins Theorem Thevenins Theorem Thevenins theorem simplifies theThvenins Theorem Thevenins Theorem Thevenins theorem simplifies the
LEMIs Theorem Lamis Theorem Statement Lamis theorem isLEMIs Theorem Lamis Theorem Statement Lamis theorem is
Bayes Theorem Basics Total probability Theorem Bayes TheoremBayes Theorem Basics Total probability Theorem Bayes Theorem