# Polynomial Division and Rational Expressions

### Topics

Factoring isn't only for when the divisor is a single term. We can use factoring to divide one polynomial by another polynomial, as long as the polynomial in the denominator is a factor of the polynomial in the numerator. We're sorry. We know you're keeping track of a lot of nomials.

For example, we can simplify by factoring 10 as 2 × 5 and canceling 2 from the top and bottom, and we can do similar things with polynomials.

When factoring, keep an eye out for common factors that are one term. No sense in making more work for yourself if you can avoid it. We know what a workaholic you are, but tone it down a bit.

### Sample Problem

Find the quotient .

The polynomial on top factors as (6*x* + 5)(6*x* – 5), so we can rewrite the expression as

.

Those all match, so now we can cancel the factor of (6*x* + 5) from the top and bottom, which leaves us with

(6*x* – 5).

### Sample Problem

What is 4*x*^{4} + 4*x*^{2} + 1 divided by 2*x* + 1?

We want to find

.

The numerator factors as (2*x*^{2} + 1)(2*x*^{2} + 1), so we can rewrite the problem as

.

Then we cancel the factor of (2*x*^{2} + 1) from the top and bottom to find 2*x*^{2} + 1.

### Sample Problem

Find the quotient .

The numerator and denominator look like something should cancel, right? Remember: we have that guarantee that the problems in this section will work out evenly, so we can take that to the bank. Actually, don't try taking that to the bank. They'll give you a weird look.

The first thing we're supposed to do is look for common factors, but there are no common factors shared by both the numerator and the denominator. They could probably learn a lot about sharing if they watched The Wiggles once in a while. Anyway, what we *can* do is pull out a common factor of 2 in the numerator to simplify that guy at least. Hopefully, doing this will shed some light on where we go next. We can now rewrite our original problem as

.

The expressions 5*x* – 1 and 1 – 5*x* are negatives of each other. We're getting closer, we can feel it already:

(1 – 5*x*) = (-1)(5*x* – 1).

If we factor out -1 from the denominator, we have

.

Now we can cancel 5*x* – 1 from the top and bottom to find

.

When asked to divide polynomials, look for cases like this where expressions in the numerator and denominator are negatives of each other. Then factor out -1 and force them to get along.

Factoring works on multivariable polynomials too. Oh, happy day. We were starting to miss our old friends *y* and *z*.