Question

How do you express `(x^3 - 5x + 2) / (x^2 - 8x + 15)` in partial fractions?

Answer 1

`(x^3-5x+2)/(x^2-8x+15) = x+8-7/(x-3)+51/(x-5)`

Explanation:

`(x^3-5x+2)/(x^2-8x+15)`

`=(x^3-8x^2+15x+8x^2-64x+120+44x-118)/(x^2-8x+15)`

`=(x(x^2-8x+15)+8(x^2-8x+15)+44x-118)/(x^2-8x+15)`

`=x+8+(44x-118)/(x^2-8x+15)`

`=x+8+(44x-118)/((x-3)(x-5))`

`=x+8+A/(x-3)+B/(x-5)`

Using Heaviside's cover-up method, we find:

`A=(44(3)-118)/((3)-5) = (132-118)/(-2) = 14/(-2) = -7`

`B=(44(5)-118)/((5)-3) = (220-118)/2 = 102/2 = 51`

So:

`(x^3-5x+2)/(x^2-8x+15) = x+8-7/(x-3)+51/(x-5)`