# 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).