# Polynomial Division and Rational Expressions

### Topics

## Introduction to :

First, you take the rubber ball in your left hand, then tap the back of that hand with the wand held in your right. Say a few magic words, and then...oh. This isn't amateur magician hour? Our bad.

We still have a good trick for you to try, though. It may not impress your baby brother or Grandma Jeannie, but it *will* impress the person grading your algebra problems, and that's a swell magic trick in its own right.

When we divided polynomials by monomials—not polynomials that have contracted a viral infection, in case you were wondering—we split up the terms of the polynomial and divided each term by the monomial. We can still split up the terms of the polynomial even if we're dividing by another polynomial, except we'll need to do it in a clever way. We also shouldn't wear mismatched socks while we do this, since it damages our cleverness rating.

### Sample Problem

Find .

Split the fraction up into two fractions. Peel it apart like you're getting ready to eat some string cheese.

Now the first fraction is only a cleverly disguised version of 1. The second fraction can't be made any nicer, since the degree of the numerator is less than the degree of the denominator. At least we're not stuck with something like Kevin Bacon^{6}.

That one can get complicated.

Because we can't do anything more with the second fraction,

.

By carefully splitting up polynomials and using the fact that adding 0 to an expression doesn't change the value of the expression (it also doesn't change the amount of our allowance, *Dad*), we arrive at another way to do polynomial division. By the way, we can also add anything equivalent to 0, such as *x* – *x* or -4*y* + 4*y*, to an expression without changing the value of the expression. This will be useful because you can bend, twist, and manipulate these terms until they are exactly what you need. They are like Silly Putty in your hands. but without that nasty after-smell.

### Sample Problem

Find .

This is almost like our first example. If only we had 4*x* instead of 3*x* on the top, we would know what to do. Maybe if we close our eyes and hope *really* hard...nope. Still there. We're starting to wonder if that ever works in real life.

We could turn 3*x* into 4*x* by adding one copy of x, but that would change the value of the expression, not to mention the fact that FedEx Kinko's prices have gotten truly outrageous. It would probably cost us $20 for that one copy. However, we can add a **clever form of 0** to get 4*x* in the numerator without changing the value of the expression. If we were any more devious, we would probably be arrested.

By putting the 3*x* and the *x* together, we can rewrite this expression as

and ta-da! Now we have something we can split up.

Since the degree of -*x* – 2 is less than the degree of *x*^{2} + 4*x*, we're done. Let's pop open a bottle of Sunny D and celebrate.

#### Exercise 1

Use the trick to find the following quotient: .

#### Exercise 2

Use the trick to find the following quotient: .

#### Exercise 3

Use the trick to find the following quotient: .

#### Exercise 4

Use the trick to find the following quotient: .

#### Exercise 5

Use the trick to find the following quotient: .