# Partial Fractions

# 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).