Polynomials have tons of similarities to integers, and throughout this section we'll be coming back a lot to the integer analogy to make our points. We'll look at polynomial division in a few different situations, each of which can be related to integer division. See? We did it already.
A lot of ideas in this section come from the section on fractions, so it's a good idea to make sure you're comfortable with that stuff before diving into this section. Slipping into your Snuggie might help you feel more comfortable as well. If anyone's willing, you may also want to think about requesting a plate of apple wedges and a Capri Sun.
We'll be using fraction notation to express polynomial division. For example:
(3x2 + 9x) divided by (5x – 2) is written as 
.
We'll treat these expressions like fractions. The number on top is still called the numerator, the number on the bottom is still called the denominator, and we still aren't allowed to divide by zero. Bummer.
To start with, we'll look at polynomial division where the divisor goes into the dividend with no remainder. As a matter of fact, we can guarantee that all the division problems in this particular unit will work out evenly. Absolutely no leftovers. Our apologies to your dog.
For polynomial division problems, there are three main things to do:
- Cancel any common monomial factors that show up in both the numerator and denominator.
- Factor all polynomials and cancel any binomial or trinomial factors that appear in both the numerator and denominator.
- Use long division.