Question

How do you express `1 / ((x+5)^2 (x-1))` in partial fractions?

Answer 1

`1/((x+5)^2(x-1)) = -1/(6(x+5)^2) - 1/(36(x+5)) + 1/(36(x-1))`

Explanation:

`1/((x+5)^2(x-1)) = A/(x+5)^2 + B/(x+5) + C/(x-1)`

Multiplying both sides of this equation by `(x+5)^2` we get:

`1/(x-1) = A+B(x+5)+(C(x+5)^2)/(x-1)`

Now let `x = -5` to find:

`A = 1/((-5)-1) = -1/6`

Multiplying both sides of the first equation by `(x-1)` we get:

`1/(x+5)^2 = (A(x-1))/(x+5)^2 + (B(x-1))/(x+5) + C`

Now let `x=1` to find:

`C = 1/((1)+5)^2 = 1/36`

Going back to the first equation and making a common denominator we find:

`1/((x+5)^2(x-1)) = A/(x+5)^2 + B/(x+5) + C/(x-1)`

`color(white)(1/((x+5)^2(x-1))) = (A(x-1)+B(x-1)(x+5)+C(x+5)^2)/((x+5)^2(x-1))`

So equating the coefficients of `x^2` in the numerator, we find:

`B = -C = -1/36`