Question

Express

`\frac{4x - 1}{ {x}^(3) - 4x}`
in partial fractions

Answer 1

4x - 1 A B C
------- = --- + ---------- + --------
x^3 - 4x x x - 2 x + 2

4x - 1 = A(x - 2)(x + 2) + Bx(x + 2) + Cx(x - 2)

Let x = 2 then 4(2) - 1 = B(2)(2+2) so 7 = 8B and B = 7/8
Let x = -2 then -8 - 1 = C-2)(-2-2) so -9 = 8C and C = -9/8
Let x = 0 then -1 = -4A and A = 1/4

So the answer is

1 7 9
---- + --------- - ----------
4x 8(x - 2) 8(x + 2)