Partial Fractions

The expression

takes three simple equations and produces one over a common denominator.

It is often useful, especially in calculus to do the reverse. The resolution into partial fractions follow these rules

- If numerator ≥ denominator then divide numerator by denominator.
- For every factor of form (x - a) in the denominator there is a partial fraction .
- For every factor of form (x - a)2 in the denominator there are partial fractions and .
- For every factor of form (x - a)3 in the denominator there are partial fractions , and and so on.
- For every quadratic factor like in the denominator there is a partial fraction .
- Repeated quadratic factors has repeated forms.
- Combinations such as in the denominator there are partial fractions and .

## Worked examples.

### Example 15. Separate into partial fractions.

Let x = -2 so 9 = -3C so C = -3

Let x = 1 so 9 = 9A so A = 1

Let x = 0 so 9 = 4A - 2B – C

Substitute for A & C

9 = 4 – 2B + 3

B = -4

So

### =