Question

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

Answer 1

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

Explanation:

To write the partial fraction decomposition, first factorize the denominator
`f(x) = 3/(x^2 - 3x) = 3/(x(x-3))`
This can then be written as `A/(x) + B/(x-3)`
Reformulating this over the common denominator gives
`(A(x-3) +Bx)/(x(x-3))`
`=((A+B)x -3A)/(x(x-3))`
Therefore `(A+B) = 0` and`-3A = 3`
`:. A = -1` and `B=1`

The original expression can be written as
`-1/x +1/(x-3)`