The Big Idea
Partial fractions are the reverse of adding fractions. When you add fractions with different denominators, you find a common denominator. In partial fractions, you're splitting one complicated fraction (with a factorised denominator) into a sum of simpler fractions, each with a single linear or squared linear factor in the denominator.
- Used to simplify complex algebraic fractions for integration and binomial expansion
- Works when the denominator can be factorised into linear or squared linear factors
- Always involves the same 5-step process: factorise → split → multiply through → substitute → write out
- Requires careful algebraic manipulation and strategic choice of substitution values
Understanding the Core Idea
What Is a Partial Fraction?
A partial fraction is one of the simpler fractions you get when you break down a complex algebraic fraction. Instead of working with:
(2x + 5) / [(x + 3)(x - 2)]
You express it as:
A/(x + 3) + B/(x - 2)
where A and B are constants you can find. This process is much easier to work with when you need to integrate or expand using the binomial theorem.
Why This Matters: Some fractions are impossible to integrate in their original form. Breaking them into partial fractions makes them integrable.
Key Definitions
Linear Factor
An expression of the form (ax + b), like (x − 2) or (3x + 1). It represents a first-degree polynomial.
Squared Linear Factor
A repeated linear factor of the form (ax + b)², like (x − 3)². This appears when the same factor occurs twice in the denominator.
Coefficient
The number that multiplies a variable. In 5x + 3, the coefficient of x is 5. This is key for the "comparing coefficients" method.
Factorise
Rewrite an expression as a product of its factors. For example, x² − 5x + 6 = (x − 2)(x − 3).
Partial Fractions with Linear Denominators
This is the foundation. When your denominator can be factorised into distinct linear factors (no repeats), you use this method.
The 5-Step Process
Step 1: Factorise the denominator
Break the denominator into a product of linear factors. If needed, also factorise the numerator.
Step 2: Split into a sum
Write your fraction as a sum of simpler fractions, one for each linear factor.
Step 3: Multiply through
Multiply both sides by the entire original denominator to clear all fractions.
Step 4: Find A, B, C, etc.
Substitute clever values of x that make factors equal to zero, or use comparing coefficients.
Step 5: Write your answer
Substitute your values back into the partial fractions form.
The Pattern
For a fraction with a denominator that factors as a product of n distinct linear factors, you get n partial fractions:
General Form (Linear Denominators)
Numerator / [(ax + b)(cx + d)(ex + f)] = A/(ax + b) + B/(cx + d) + C/(ex + f)
where the numerator is usually a polynomial of degree one less than the denominator.
Worked Example: Linear Denominators
Express 3x − 2 / (5x² + 3x − 2) as partial fractions
1
2
Split into a sum