Alg5.1.12

Write $\frac{x+1}{(x-1)^2(x-2)}\,$ as partial fractions.

Let $\frac{x+1}{(x-1)^2(x-2)}=\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C}{x-2}\,$

Simplifying

$x+1=A(x-1)(x-2)+B(x-2)+C(x-1)^2\,$

Substituting $x=1\,$

$2=-B\,$

Substituting $x=2\,$

$3=C\,$

Comparing the coefficients of $x^2\,$

$0=A+C,A=-3\,$

Therefore $\frac{x+1}{(x-1)^2(x-2)}=-\frac{3}{x-1}-\frac{2}{(x-1)^2}+\frac{3}{x-2}\,$