5.5 Define Theorems and give an example
Intro to College Algebra 5.5
Zeros of Polynomial Functions Name ______________________________
Remainder Theorem:
Description: _______________________________________________________________________
_________________________________________________________________________________
Example: Evaluate f(x) = 2x5 – 3x4 – 9x3 + 8x2 + 2 at x = -3
Factor Theorem:
Description: _______________________________________________________________________
_________________________________________________________________________________
Example: Find zeros of f(x) = x3 + 4x2 – 4x – 16 given (x – 2) is a factor.
Rational Zero Theorem:
Description: _______________________________________________________________________
_________________________________________________________________________________
Example: Find rational zeros of f(x) = x3 - 5x2 + 2x + 1.
Fundamental Theorem of Algebra:
Description: _______________________________________________________________________
_________________________________________________________________________________
Example: Find zeros of f(x) = 2x3 + 5x2 - 11x + 4.
Complex Conjugate Theorem:
Description: _______________________________________________________________________
_________________________________________________________________________________
Linear Factorization Theorem:
Description: _______________________________________________________________________
_________________________________________________________________________________
Example: Find a third degree polynomial that has zeros of 5, -2i, such that f(1) = 10.
Descartes’ Rule of Signs:
Description: _______________________________________________________________________
_________________________________________________________________________________
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.
