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.