Introduction to Partial Fraction Expansion/Decomposition (PFE)

by Dr. Carol JVF Burns (website creator)
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You already know how to add fractions.
For example: $$ \cssId{s3}{\frac{1}{x-1} + \frac{3}{2x+1} \ =\ \frac{(2x+1) + 3(x-1)}{(x-1)(2x+1)} \ =\ \frac{5x-2}{2x^2-x-1}} $$ The process of going backwards:

from   $\displaystyle \frac{5x-2}{2x^2-x-1}$         back to   $\displaystyle\frac 1{x-1} + \frac{3}{2x+1}$

is called either Partial Fraction Expansion (PFE) or Partial Fraction Decomposition.

Note that adding fractions takes you from two or more fractions to a single, more complicated fraction.
Partial Fraction Expansion, on the other hand, takes you from a single (complicated) fraction to two or more simpler pieces.

This section introduces partial fraction expansion, with review of needed concepts and a simple example.
There is more detail in subsequent sections.

Which name: ‘Partial Fraction Expansion’ or ‘Partial Fraction Decomposition’?

Going from (say) $\,\color{green}{\displaystyle \frac{5x-2}{2x^2-x-1}}\,$ to the new name $\,\color{red}{\displaystyle\frac 1{x-1} + \frac{3}{2x+1}}\,$:

So, both names are appropriate.
Since ‘expansion’ seems a bit more optimistic than ‘decomposition’,
this author prefers the name Partial Fraction Expansion, and usually abbreviates it as PFE.

What fractions can PFE be used for?

Recall that a rational function is a ratio of polynomials (with a nonzero denominator).
That is, a rational function is a fraction, with any polynomial in the numerator, and a nonzero polynomial in the denominator.

Theoretically, PFE can be used on any rational function.
(However, the second step below limits its usefulness, in practice.)

Which direction is easier: Adding Fractions or Partial Fraction Expansion?

Adding fractions is a lot easier than PFE!

Partial Fraction Expansion involves:

The example below shows how to go from $\,\displaystyle \frac{5x-2}{2x^2-x-1}\,$ to $\,\displaystyle\frac 1{x-1} + \frac{3}{2x+1}\,.$

Even in this simple case, you'll see that it's a bit of work.

Uses for Partial Fraction Expansion

PFE is valuable whenever you need to represent a complicated fraction of polynomials as a sum of simpler fractions.
Two areas where PFE is frequently used:

(Don't worry if you have no idea what these applications are—yet!)

Review of Concepts and Terminology Needed for PFE

Partial Fraction Expansion draws on lots of beautiful mathematical theory!

A Simple Example: PFE with Distinct Linear Factors

This simple example shows the basic process of Partial Fraction Expansion.
The way to handle situations other than distinct linear factors is discussed in subsequent sections.

Find the partial fraction expansion of $\,\displaystyle\frac{5x-2}{2x^2 - x - 1}\,$
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Partial Fraction Expansion: Linear Factors

On this exercise, you will not key in your answer.
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