# Partial Fractions, Long Division

A **rational expression **is a quotient of polynomials, i.e. has the form

At first sight, rational expressions are difficult to integrate. However, the

algebraic techniques of partial fraction expansion and long division can be used

to make these complex expressions simpler.

**1.1 Partial fraction expansion**

Word of Warning: Our method here applies when the denominator has linear,

nonrepeated factors only. This will be sufficient for our Calc II class, but
some

of you will see more sophisticated methods in other classes.

Partial fractionsworks when the largest power of the numerator is smaller

than the largest power of the denominator. Here are the instructions. Start

with a rational expression:

**First,** completely factor the bottom polynomial:

**Second**, break up the fraction so that the factors of the bottom become
the

denominators of individual fractions, with (as of yet) unknown constants in

the numerators:

**Third**, figure out what the constants
are.

**Example 1** Use partial fraction expansion to simplify

**Solution** We simply follow the instructions laid out above:

First, factor the bottom:

Second, break up the fraction

Third, figure out what A_{1} and A_{0} are:

Thus x = (A_{1} +A_{0})x + 2A_{1} + 3A_{0}, so that A_{1} +A_{0} = 1 and 2A_{1} +3A_{0} = 0.

We can solve these two equations, to get A_{1} = 3, A_{0} = -2. Thus
finally

**Example 2** Evaluate

**Solution** The fraction is too tough to evaluate, so we have to use
partial fraction

expansion to simplify it.

Thus A_{1} + A_{0} = 2 and -3A_{1}
- 4A_{0} = -1. We can solve this to get A_{1} = 7,

A_{0} = -5. Therefore

Now we can solve our calculus problem:

**1.2 Long Division**

Long division (or synthetic division) can be used when the highest power on

top is equal to or larger than the highest power on bottom.

**Example 3** Evaluate

Solution The fraction is too tough to evaluate directly. Since the top power
is

bigger, we have to use long division. Using the long division process (which

we discussed in class), we get

Note: you can also use synthetic division to arrive at this conclusion. Now
we

can solve our calculus problem: