Complex Zeros and the Fundamental Theorem of Algebra

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Presentation transcript:

Complex Zeros and the Fundamental Theorem of Algebra 2.5 Complex Zeros and the Fundamental Theorem of Algebra

Quick Review

Quick Review Solutions

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.

Fundamental Theorem of Algebra A polynomial function of degree n has n complex zeros (real and nonreal). Some of these zeros may be repeated.

Linear Factorization Theorem

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). NOTE: k is not an x-intercept

Example 1 Exploring Fundamental Polynomial Connections

Example 1 Exploring Fundamental Polynomial Connections

Complex Conjugate Zeros

Example 2 Finding a Polynomial from Given Zeros

Example 4 – Factoring a Polynomial with Complex Zeros

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.