Partial Fractions

Definition

Partial Fractions refer to the method used to split complex fractions into simpler, separate fractions.

When to Use

Partial fractions is typically used when you need to integrate or differentiate a rational function, or when simplifying complex fractions in algebraic manipulations.

Types of Partial Fractions

Depending on the form of the denominator, there are three types of partial fraction decomposition: Proper Rational Function, Improper Rational Function, and Rational Function with Repeated Roots.

Proper Rational Function

A Proper Rational Function is a rational function where the degree of the numerator is less than the degree of the denominator.
For instance, if the denominator as (ax + b)(cx + d), the fraction can be expressed as A/(ax + b) + B/(cx + d), where A and B are constants.

Improper Rational Function

An Improper Rational Function occurs when the degree of the numerator is equal to or higher than the degree of the denominator.
It’s necessary to use polynomial division before proceeding with the partial fractions decomposition.

Rational Function with Repeated Roots

A Rational Function with Repeated Roots involves denominators with repeated linear factors.
A separate fraction is needed for each term of the repeated root in the denominator.

How to Decompose

To decompose a rational function into partial fractions, first factorise the denominator completely, and then assign a different constant to the numerator of each fraction.
Construct simultaneous equations by equating coefficients of the equivalent polynomial expressions, and then solve for the constants.

Key Method Steps

Step 1: Make sure that the fraction is proper. If it isn’t, use polynomial division.

Step 2: Factorise the denominator completely.

Step 3: For each factor in the denominator, write down a fraction with the corresponding factor in the denominator and an undefined constant in the numerator.

Step 4: Create equations by equating the original rational function to the sum of the fractions you have just written down.

Step 5: Solve these equations simultaneously to find the constants in the numerators.

Integration

If a function which you want to integrate can be expressed as a sum of partial fractions, the integration becomes much easier as you can integrate each simple fraction separately.
Remember to use logarithms when integrating denominators of the form (ax + b).