# 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

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

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 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.

1. Use synthetic division to divide the polynomial by
$\left(x-k\right)$
.
2. Confirm that the remainder is 0.
3. Write the polynomial as the product of
$\left(x-k\right)$
4. If possible, factor the quadratic.
5. 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:

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

We can factor the quadratic factor to write the polynomial as

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

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