Polynomial Functions Lesson 9.2

Polynomials Definition:  The sum of one or more power function  Each power is a non negative integer

Polynomials General formula  a 0, a 1, …,a n are constant coefficients  n is the degree of the polynomial  Standard form is for descending powers of x  a n x n is said to be the “leading term”

Polynomial Properties Consider what happens when x gets very large negative or positive  Called “end behavior”  Also “long-run” behavior Basically the leading term a n x n takes over Compare f(x) = x 3 with g(x) = x 3 + x 2  Look at tables  Use standard zoom, then zoom out

Polynomial Properties Compare tables for low, high values

Polynomial Properties Compare graphs ( -10 < x < 10) For 0 < x < 500 the graphs are essentially the same The leading term x 3 takes over

Zeros of Polynomials We seek values of x for which p(x) = 0 Consider  What is the end behavior?  What is q(0) = ?  How does this tell us that we can expect at least two roots?

Methods for Finding Zeros Graph and ask for x-axis intercepts Use solve(y1(x)=0,x) Use zeros(y1(x)) When complex roots exist, use cSolve() or cZeros()

Practice Given y = (x + 4)(2x – 3)(5 – x)  What is the degree?  How many terms does it have?  What is the long run behavior? f(x) = x 3 +x + 1 is invertible (has an inverse)  How can you tell?  Find f(0.5) and f -1 (0.5)

Assignment Lesson 9.2 Page 400 Exercises 1 – 29 odd