If we multiply the numerators and denominators together, we'll find ourselves in a horrible mess and won't be able to do anything. If we factor first, the problem turns into a much less nauseating beast in which several factors can cancel out. Yeah, that second one sounds good.

Example 2

Multiply .

The first thing we do is factor. Then, the problem looks like this:

When we do the multiplication, lots of junk cancels out:

Our final answer is , which isn't that horrible at all. One might even say it's kind of pleasant. Just kidding! (Not kidding.)

Example 3

Find .

First, we factor:

Then we write the polynomial as a fraction by introducing it to a denominator of 1. ("Polynomial, 1, 1, Polynomial. Oh, you've met.")

We change division to multiplication by the reciprocal and do any possible simplifying. Luckily, there's not much in this case.