Question

How do you express `( x-2 ) / (x^2 + 4x + 3)` in partial fractions?

Answer 1

`5/(2(x+3)) - 3/(2(x+1))`

Explanation:

The first step here is to factor the denominator

`x^2 + 4x + 3 = (x+1)(x+3)`

since these factors are linear the numerator will be a constant

`(x-2)/((x+1)(x+3) ) = A/(x+1) + B/(x+3)`

the next step is to multiply both sides by (x+1)(x+3)

hence x - 2 = A(x+3) + B(x+1)

Note that if x = - 3 and x = -1 then the terms with A and B will be zero

let x = - 3 : - 5 = -2B → B`= 5/2`

let x = - 1 : - 3 = 2A → A #= -3/2

`rArr (x-2)/(x^2+4x+3) = 5/(2(x+3)) - 3/(2(x+1))`