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

##### 1 Answer

Jan 15, 2016

#( -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 :