Partial fractions
Partial fractions split one algebraic fraction into a sum of simpler ones. It is a stepping stone to integrating fractions and to binomial expansions with fractional powers.
Part 1 of 3: Learn it
In short
- Distinct linear factors: one fraction per factor, A/(x − a) + B/(x − b).
- A repeated factor (x − a)² needs A/(x − a) + B/(x − a)².
- Multiply through by the denominator, then substitute convenient values of x.
Where this is in your specification
Spec points: DfE B8 (AQA 7357 B8, Edexcel 9MA0 Pure topic 2, OCR A H240 1.02)
| Board | Topic: Polynomials and partial fractions |
|---|---|
| DfE content | B7, B8 |
| AQA 7357 | B7, B8 |
| Edexcel 9MA0 | Pure topic 2 |
| OCR H240 | 1.02 |
| Higher C847 76 | Algebraic and trigonometric skills: polynomials, synthetic division |
Setting up
| Denominator | Partial fraction form |
|---|---|
| (x − a)(x − b) | A/(x − a) + B/(x − b) |
| (x − a)(x − b)(x − c) | A/(x − a) + B/(x − b) + C/(x − c) |
| (x − a)(x − b)² | A/(x − a) + B/(x − b) + C/(x − b)² |
This works when the top has a lower degree than the bottom. If not, divide first.
Finding the constants
- Multiply both sides by the full denominator so there are no fractions.
- Substitute the value of x that makes each factor zero: each one picks out a constant.
- If a constant is left (from a repeated factor), compare the coefficients of x² or the constant terms.
Express 4/((x − 1)(x + 3)) in partial fractions.
Show the answer
1/(x − 1) − 1/(x + 3).
Part 2 of 3: See it worked
Worked examples
Example 1
Split (8x + 1)/((x − 1)(x + 2)) into partial fractions.
- 8x + 1 = A(x + 2) + B(x − 1)
- x = 1: 9 = 3A, so A = 3
- x = −2: −15 = −3B, so B = 5
Answer: 3/(x − 1) + 5/(x + 2).
Example 2
Express (3x² + 9x − 3)/((x − 1)(x + 2)²) in partial fractions.
- 3x² + 9x − 3 = A(x + 2)² + B(x − 1)(x + 2) + C(x − 1)
- x = 1: 9 = 9A, so A = 1. x = −2: −9 = −3C, so C = 3
- Compare x² terms: 3 = A + B, so B = 2
Answer: 1/(x − 1) + 2/(x + 2) + 3/(x + 2)².
Common mistakes
- Leaving out the A/(x − b) term when (x − b) is repeated.
- Making sign errors when substituting a negative value of x.
- Forgetting to divide first when the top has the same or a higher degree than the bottom.
- Not checking the answer by recombining or substituting a value.
What form is used for a denominator of x(x + 5)²?
Show the answer
A/x + B/(x + 5) + C/(x + 5)².
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Express 4/((x − 1)(x + 3)) in partial fractions.
1/(x − 1) − 1/(x + 3).
What form is used for a denominator of x(x + 5)²?
A/x + B/(x + 5) + C/(x + 5)².
Why choose x = 1 when the denominator contains (x − 1)?
It makes that factor zero, so the other terms vanish and one constant can be found directly.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
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