# 5.5 Define Theorems and give an example

Intro to College Algebra 5.5

Zeros of Polynomial Functions                      Name ______________________________

Remainder Theorem:

Description: _______________________________________________________________________

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Example:   Evaluate f(x) = 2x5 – 3x4 – 9x3 + 8x2 + 2 at x = -3

Factor Theorem:

Description: _______________________________________________________________________

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Example:   Find zeros of f(x) = x3 + 4x2 – 4x – 16  given (x – 2) is a factor.

Rational Zero Theorem:

Description: _______________________________________________________________________

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Example: Find rational zeros of  f(x) = x3 - 5x2 + 2x + 1.

Fundamental Theorem of Algebra:

Description: _______________________________________________________________________

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Example:  Find zeros of  f(x) = 2x3 + 5x2 - 11x + 4.

Complex Conjugate Theorem:

Description: _______________________________________________________________________

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Linear Factorization Theorem:

Description: _______________________________________________________________________

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Example:    Find a third degree polynomial that has zeros of 5, -2i, such that f(1) = 10.

Descartes’ Rule of Signs:

Description: _______________________________________________________________________

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Example: Use Descartes’ Rule of Signs to find the maximum number of positive and negative real zeros for f(x) =  2x4 – 10x3 + 11x2 – 15x + 12.  Sketch a graph to verify.