Question

How do you write the partial fraction decomposition of the rational expression `(x^3 - 6x^2 + 11x - 6) / (4x^3 - 28x^2 + 56x - 32)`?

Answer 1

`(x^3-6x^2+11x-6)/(4x^3-28x^2+56x-32)=1/4+1/(4(x-4))`

with exclusions `x != 1` and `x != 2`

Explanation:

Let us factor the numerator and denominator first:

`x^3-6x^2+11x-6`

`=(x-1)(x^2-5x+6)`

`=(x-1)(x-2)(x-3)`

`color(white)()`

`4x^3-28x^2+56x-32`

`=4(x^3-7x^2+14x-8)`

`=4(x-1)(x^2-6x+8)`

`=4(x-1)(x-2)(x-4)`

So:

`(x^3-6x^2+11x-6)/(4x^3-28x^2+56x-32)`

`=(color(red)(cancel(color(black)((x-1))))color(red)(cancel(color(black)((x-2))))(x-3))/(4color(red)(cancel(color(black)((x-1))))color(red)(cancel(color(black)((x-2))))(x-4))`

`=(x-3)/(4(x-4))`

`=(x-4+1)/(4(x-4))`

`=1/4+1/(4(x-4))`

with exclusions `x != 1` and `x != 2`