Question

How do you write the partial fraction decomposition of the rational expression `(x+4)/((x+1)(x-2)^2 )`?

Answer 1

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

Explanation:

firstly note that `x-2)^2`
has factors (x-2) and `(x-2)^2`

the factors of the denominator are linear hence the numerators are constants.

`rArr (x+4)/((x+1)(x-2)^2 ) = A/(x+1) + B/(x-2) + C/(x-2)^2`

now multiply through by `(x+1)(x-2)^2`

hence : x+4 = `A(x-2)^2 + B(x+1)(x-2) + C(x+1)....................(1)`

the aim now is to find values for A , B and C . Note that if x = 2 then the terms with A and B will be zero , and if x = -1 the terms with B and C will be zero. This is the starting point in finding values.

let x = 2 in (1) : `6 = 3C rArr C = 2`

let x = -1 in (1) : -3 =`9A rArr A = 1/3`
any value for x may be chosen to find B.

let x = 0 in (1) : 4 = 4A - 2B + C `rArr 2B = 4A + C - 4`

hence 2B `= 4/3 + 2 - 4 = -2/3 rArr B = -1/3`

`rArr(x+4)/((x+1)(x-2)^2) = 1/3(x+1) -1/3(x-2) + 2/(x-2)^2`