The Real Zeros of a Polynomial Function Obj: Apply Factor Theorem, Use Rational Zero Theorem to list roots, Apply Descartes’ Rule of Signs to determine.

The Real Zeros of a Polynomial Function

Obj: Apply Factor Theorem, Use Rational Zero Theorem to list roots, Apply Descartes’ Rule of Signs to determine possible positive and negative real zeros

Why?

We can easily find the zeros of a quadratic by factoring or using the Quadratic Formula. In this section we will learn more about the real zeros of a polynomial and in 4.5 we will learn about the complex zeros of a polynomial.

In 4.2 it was easy to find the zeros because the polynomials given were easily factorable. In the next two sections, we will find the zeros when factoring is not as apparent….

Use the Factor Theorem

Show that x – 3 is a factor of x3 + 4x2 – 15x – 18. If so, factor the polynomial completely.

The Number of Real Zeros

A polynomial function cannot have more real zeros than its degree.

Not all zeros of a polynomial are real, either! (we will see this in 4.5)

Example: The following functions illustrate that a polynomial function of degree n can have at most n real zeros.

P(x) Degree Real Zeros Comments

f(x) = x2 – 9

f(x) = x2 + 4

f(x) = x3 – 1

f(x) = x3 – x2 – 6x

Rational Zero Theorem

n

0

a of Factors

a of Factors ZerosPossible

B. List all of the possible rational zeros of f(x) = x3 + 3x + 24.

Identify Possible Zeros

List all of the possible rational zeros of f(x) = 3x4 – x3 + 4.

Descartes’ Rule of Signs

State the possible number of positive real zeros, negative real zeros of p(x) = –x6 + 4x3 – 2x2 – x – 1.

Use Descartes’ Rule of Signs to determine the number and type of real zeros. Count the number of changes in sign for the coefficients of p(x).

Find Numbers of Positive and Negative Zeros

p(x) = –x6 + 4x3 – 2x2 – x –1

yes– to +

yes+ to –

no– to –

no– to –2 or 0 positive real

zeros

Since there are two sign changes, there are 2 or 0 positive real zeros. Find p(–x) and count the number of sign changes for its coefficients.

Find Numbers of Positive and Negative Zeros

p(–x) = –(–x)6 + 4(–x)3 – 2(–x)2 – (–x) –1

Find all of the zeros of f(x) = x3 – x2 + 2x + 4.