1EPartialFracs

​ **Partial Fractions**

Recall that to add two fractions, first they must be changed to have a common denominator. For example, math . \qquad \dfrac {3}{5} + \dfrac {1}{4} = \dfrac {12}{20} + \dfrac {5}{20} = \dfrac {17}{20} math

Aim: To do the reverse of that process ie split one fraction into two. The resulting two fractions are called __partial fractions__.

See Introduction to Partial Fractions

There are four types of question you will encounter. Each one has an almost identical approach except for the first step.

Type 1. Linear numerator, quadratic denominator with two factors

math \text {Separate } \dfrac {2x+1}{(x-2)(x-3)} \text { into partial fractions.} math

Divide into two fractions with numerators A and B (constants)

math \text {Let } \dfrac {2x+1}{(x-2)(x-3)} = \dfrac {A}{x-2} + \dfrac {B}{x-3} math etc

See Example Type 1

Type 2. Numerator is polynomial of degree ≥ denominator

math \text {Express } \dfrac {3x^2 + 3x + 4}{(x-2)(x-3)} \text { into partial fractions.} math

First expand denominator and do long division, then do __partial fractions__ on the rational term etc

See Example Type 2

Type 3 Denominator is a Perfect Square

math \text {Express } \dfrac {2x - 1}{(x+3)^2} \text { into partial fractions.} math

The two denominators should be the single term and the squared term

math \text {Let } \dfrac {2x-1}{(x+3)^2} = \dfrac {A}{x+3} + \dfrac {B}{(x+3)^2} math etc

See Example Type 3

Type 4 Quadratic in denominator can’t be factorised

math \text {Express } \dfrac {x+3}{(x+2)(x^2+1)} \text { in partial fractions} math

Because x 2 + 1 can't be factorised over R, use a polynomial (for the numerator) of one degree less than the denominator.

math \dfrac {x+3}{(x+2)(x^2+1)} = \dfrac {A}{x+2} + \dfrac {Bx + C}{x^2+1} math etc

See Example Type 4

Note that these approaches can be combined. Simply add as many fraction terms of the appropriate type on the right hand side. Return to Summary of Coordinate Geometry

.