Question

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

Answer 1

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

Explanation:

`(x+2)/(x(x-4))`

`(x+2)/(x(x-4)) = A/x + B/(x-4)`

`(x+2)/(x(x-4)) =(A(x-4)+B(x))/(x(x-4))`

`(x+2) = A(x-4)+B(x)`

Let `x=0` This done to remove the `B`
`(0+2) = A(0-4)+B(0)`

`2=-4A`

`2/-4 =(-4A)/-4`
`-1/2 = A`

`A=-1/2`

Now let `x=4` Which makes `(x-4)` as zero and thus eliminating `A` and we can solve for `B`

`(4+2)=A(4-4)+B(4)`
`6=4B`
`6/4 = (4B)/4`
`3/2 = B`

`B=3/2`

Therefore,

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

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