Question

How do you write the partial fraction decomposition of the rational expression `(x^2 - 3x) / ((x-1)(x+2))`?

Answer 1

`( -2/3)/(x - 1 ) -(10/3)/(x + 2 )`

Explanation:

let `( x^2 - 3x) /((x - 1 )(x + 2 )) ≣ A/(x - 1) + B/(x + 2 )`

Since the factors on the denominator are of degree 1 (linear) then the numerators will be constants (degree 0 ) denoted by A and B .

Multiply both sides of the equation by (x - 1 )(x + 2 ) :

`rArr x^2 - 3x = A( x + 2 ) + B (x - 1 )..................color(red)((*))`

(Note that if x = 1 then the term in B will be 0. Similarly if x = - 2 then the term in A will also be 0 . )

let x = 1 and substitute in equation `color(red)((*))`

`rArr - 2 = 3A rArr A = - 2/3`

let x = -2 and substitute in equation `color(red)((*))`

`rArr 10 = - 3B rArr B = -10/3`

Finally :

`( x^2 - 3x)/((x - 1 )(x +2 )) = (-2/3)/(x - 1 ) -(10/3 )/(x - 2 )`