# The Method of Partial Fractions

The Fundamental Theorem of Algebra thus tells us that there are 4 different "simplest'' denominator types:

• linear factors,
• irreducible factors of degree 2,
• repeated linear factors, and
• repeated irreducible factors of degree 2.
The method of partial fractions deals with each of them in a slightly different manner.

• If you encounter a linear factor (x-a) (as (x-4) in

you will use a summand of the form

• An irreducible quadratic term of the form (as in

will contribute two summands of the form

Thus--using different variable names for each term-- the expression above will take the form:

The order in which you write down the terms does not matter, you could have written

• A repeated linear factor will contribute n terms of the form

For instance

• Last not least, a repeated irreducible factor of degree 2 of the form will contribute 2n terms of the form:

#### A Hint.

The number of letters you use always equals the degree of the denominator polynomial! This rule does not only apply "globally'', but also "locally'': A repeated linear factor as yields four expressions with four variable letters.

#### Try it yourself!

Predict the form of the right hand side for the following rational functions: