Section 3.3

Real Zeros of Polynomial Functions

Real Zeros of Polynomial Functions

Objectives:– Use synthetic and long division– Use the Remainder and Factor Theorem– Use the Rational Zeros Theorem– Find the Real Zeros of a Polynomial Function– Solve Polynomial Equations– Use the Theorem for Bounds on Zeros– Use the Intermediate Value Theorem

Synthetic and Long Division of Polynomials

EXAMPLES

Remainder Theorem

Let f be a polynomial function. If f(x) is divided by x-c, then the remainder is f(c).

Find the remainder if

is divided by

What is f(-3)?

3 2( ) 3 9 18 24f x x x x 3x

Factor Theorem

Let f be a polynomial function. Then x-c is a factor of f(x) if and only if f(c)=0. So this means that the remainder when the polynomial is divided by x-c is 0. Thus x-c divides into the polynomial evenly.

Theorem: A polynomial function of degree n has at most n real zeros.

Rational Zeros Theorem

Let f be a polynomial function of degree 1 or higher of the form

Where each coefficient is an integer. If p/q, in lowest terms, is a rational zero of f, then p must be a factor of and q must be a factor of .

We can test each possible solution with synthetic division.

0a na

11 1 0( ) ...n n

n nf x a x a x a x a

Bounds on Zeros

f(x) is a polynomial function whose leading coefficient is 1

A bound is the smaller of the following two numbers:

OR…

Write the bound as plus or minus.

0 1 2 1Max{1,abs( ) abs( ) abs( ) ... abs( )}na a a a

0 1 2 11 Max{abs( ),abs( ),abs( ),...abs( )}na a a a

Finding Zeros of Polynomials

Use the Rational Zeros Theorem and repeated division to find the zeros

EXAMPLES