### CHECK OUT SHMOOP'S FREE STUDY TOOLS:

# Polynomial Division and Rational Expressions

### CHECK OUT SHMOOP'S FREE STUDY TOOLS:

# Polynomial Division and Rational Expressions Examples

#### Common Factors

The easiest case of polynomial division is when the divisor is a single term that happens to be a common factor of the dividend. You'll love when this happens. For the sake of the integer analogy,...

#### Factoring

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 poly...

#### Long Division

After looking for single-term factors and finishing all our factoring, the next thing we do to divide polynomials is use good old-fashioned long division. You know, old-fashioned like the donut: de...

#### Common Factors

The first thing to do when dividing polynomials is to look for and remove common factors. In terms of rational expressions, "looking for common factors" means that we're looking for those that go i...

#### Long Division

Once we've canceled all the factors we can, we write out the new numerator and denominator (the unfactored versions) and look at their degrees. If it's too many degrees, we'll stay inside and eat c...

#### A Clever Trick

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.W...

#### Simplifying Rational Expressions

When we're given a rational expression, we can simplify it by doing the first two steps of polynomial division: we cancel any common factors, then factor the numerator and denominator and cancel an...

#### Multiplying and Dividing Rational Expressions

MultiplyingAs you'll recall from that time you became the grandmaster lord high poobah of all things fractions, multiplying fractions requires us to multiply the numerator by the numerator and the...

#### Adding and Subtracting Rational Expressions

...with the same denominator
Adding or subtracting rational expressions with the same denominator is like adding or subtracting fractions with the same denominator. We add or subtract t...

#### Simplifying Complex Rational Expressions

A complex rational expression is a rational expression in which the numerator and/or the denominator are rational expressions. In other words, a complex rational expression is a nasty-looking ratio...

#### Equations Involving Rational Expressions

Solving an equation that contains rational expressions is similar to solving any old equation that has fractions in it. In fact, if the variable only occurs in numerators, we are actually solving a...