Question

partial fraction x/(x-1) (x-2)

Answer 1

Let
$\frac{x}{(x-1)(x-2)}=\frac{A}{(x-1)}+\frac{B}{(x-2)}$
for finding value of A ,

$\frac{x}{(x-1)(x-2)}\times\:(x-1)=\frac{A}{(x-1)}\times\:(x-1)\\\\\frac{x}{(x-2)}=A_{x=1}$
$A = \frac{1}{-1}=-1$

again, for finding value of B ,
$\frac{x}{(x-1)(x-2)}\times\:(x-2)=\frac{B}{(x-1)(x-2)}\times\:(x-1)$
$B_{x=2}=\frac{2}{2-1}=2$

hence,
$\frac{x}{(x-1)(x-2)}=\frac{-1}{(x-1)}+\frac{2}{(x-2)}$