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Algebra - Notes
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## Partial Fractions

This section doesn’t really have a lot to do with the rest of this chapter, but since the subject needs to be covered and this was a fairly short chapter it seemed like as good a place as any to put it.

So, let’s start with the following.  Let’s suppose that we want to add the following two rational expressions.

What we want to do in this section is to start with rational expressions and ask what simpler rational expressions did we add and/or subtract to get the original expression.  The process of doing this is called partial fractions and the result is often called the partial fraction decomposition.

The process can be a little long and on occasion messy, but it is actually fairly simple. We will start by trying to determine the partial fraction decomposition of,

where both P(x) and Q(x) are polynomials and the degree of P(x) is smaller than the degree of Q(x).   Partial fractions can only be done if the degree of the numerator is strictly less than the degree of the denominator.  That is important to remember.

So, once we’ve determined that partial fractions can be done we factor the denominator as completely as possible.  Then for each factor in the denominator we can use the following table to determine the term(s) we pick up in the partial fraction decomposition.

 Factor in denominator Term in partial fraction decomposition

Notice that the first and third cases are really special cases of the second and fourth cases respectively if we let .  Also, it will always be possible to factor any polynomial down into a product of linear factors ( ) and quadratic factors ( ) some of which may be raised to a power.

There are several methods for determining the coefficients for each term and we will go over each of those as we work the examples.  Speaking of which, let’s get started on some examples.