Three Techniques for Evaluating and Finding Zeros of Polynomial Functions

The Factor Theorem is another theorem that helps us analyze polynomial equations. It tells us how the zeros of a polynomial are related to the factors. Recall that the Division Algorithm.

$f(x)=(x-k) q(x)+r$

If $k$ is a zero, then the remainder $r$ is $f(k)=0$ and $f(x)=(x−k)q(x)+0$ or $f(x)=(x−k)q(x)$.

Notice, written in this form, $x−k$ is a factor of $f(x)$. We can conclude if $k$ is a zero of $f(x)$, then $x−k$ is a factor of $f(x)$.

Similarly, if $x−k$ is a factor of $f(x)$, then the remainder of the Division Algorithm $f(x)=(x−k)q(x)+r$ is $0$. This tells us that $k$ is a zero.

This pair of implications is the Factor Theorem. As we will soon see, a polynomial of degree n in the complex number system will have n zeros. We can use the Factor Theorem to completely factor a polynomial into the product of n factors. Once the polynomial has been completely factored, we can easily determine the zeros of the polynomial.

THE FACTOR THEOREM

According to the Factor Theorem, $k$ is a zero of $f(x)$ if and only if $(x−k)$ is a factor of $f(x)$.

HOW TO

Given a factor and a third-degree polynomial, use the Factor Theorem to factor the polynomial.

EXAMPLE 2

Using the Factor Theorem to Find the Zeros of a Polynomial Expression

Show that $(x+2)$ is a factor of $x^{3}-6 x^{2}-x+30$. Find the remaining factors. Use the factors to determine the zeros of the polynomial.

Solution

We can use synthetic division to show that $(x+2)$ is a factor of the polynomial.

The remainder is zero, so $(x+2)$ is a factor of the polynomial. We can use the Division Algorithm to write the polynomial as the product of the divisor and the quotient:

$(x+2)\left(x^{2}-8 x+15\right)$

We can factor the quadratic factor to write the polynomial as

$(x+2)(x−3)(x−5)$

By the Factor Theorem, the zeros of $x^{3}-6 x^{2}-x+30$ are $–2$, $3$, and $5$.

TRY IT #2

Use the Factor Theorem to find the zeros of $f(x)=x^{3}+4 x^{2}-4 x-16$ given that $(x−2)$ is a factor of the polynomial.