Use the Factor Theorem to solve a polynomial equation

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 tells us

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

$f\left(x\right)=\left(x-k\right)q\left(x\right)+r$ .

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

Notice, written in this form, x – k is a factor of $f\left(x\right)$ . We can conclude if k is a zero of $f\left(x\right)$ , then $x-k$ is a factor of $f\left(x\right)$ .

Similarly, if $x-k$ is a factor of $f\left(x\right)$ , then the remainder of the Division Algorithm $f\left(x\right)=\left(x-k\right)q\left(x\right)+r$ 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.

A General Note: The Factor Theorem

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

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

Use synthetic division to divide the polynomial by $\left(x-k\right)$ .
Confirm that the remainder is 0.
Write the polynomial as the product of $\left(x-k\right)$ and the quadratic quotient.
If possible, factor the quadratic.
Write the polynomial as the product of factors.

Example 2: Using the Factor Theorem to Solve a Polynomial Equation

Show that $\left(x+2\right)$ 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.

Solutions

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

The remainder is zero, so $\left(x+2\right)$ 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) (x^{2} - 8 x + 15)

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

We can factor the quadratic factor to write the polynomial as

(x + 2) (x - 3) (x - 5)

$\left(x+2\right)\left(x - 3\right)\left(x - 5\right)$

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 $f\left(x\right)={x}^{3}+4{x}^{2}-4x - 16$ given that $\left(x - 2\right)$ is a factor of the polynomial.

Solution

Zeros of Polynomial Functions