Polynomial Functions

Lesson 9.2

Polynomials

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

3 2 4( ) 5 7 123

f x x x x

Polynomials

• General formula

a0, a1, … ,an are constant coefficients n is the degree of the polynomial Standard form is for descending powers of x anxn is said to be the “leading term”

11 1 0( ) ...n n

n nP x a x a x a x a

Polynomial Properties

• Consider what happens when x gets very large negative or positive Called “end behavior” Also “long-run” behavior

• Basically the leading term anxn takes over

• Comparef(x) = x3 with g(x) = x3 + x2

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 < 500the graphs are essentially the same

The leading term x3 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?

6 5 2( ) 3 2 4 1q x x x x

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

• Giveny = (x + 4)(2x – 3)(5 – x) What is the degree? How many terms does it have? What is the long run behavior?

• f(x) = x3 +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