Question

Resolve $\frac{x^2-5x-1}{(x-1{)}^2(x-2)}$ into partial fraction.

Answer 1

$\frac{x^2-5x-1}{(x-1{)}^2(x-2)}=\frac{A}{x-1}+\frac{B}{(x-1{)}^2}+\frac{C}{x-2}$
$\Rightarrow \frac{x^2-5x-1}{(x-1{)}^2(x-2)}=\frac{A(x-2)(x-1)+B(x-2)+C(x-1{)}^2}{(x-1{)}^2(x-2)}.....\left(2\right)$
$\Rightarrow x^2-5x-1=A(x-2)(x-1)+B(x-2)+C(x-1{)}^2.....(1)$
Putting $x-1=0\Rightarrow x=1$ in equation (2) we get.
$\Rightarrow 1^2-5\times 1-1=0+B(1-2)+0$
$\Rightarrow -5=-B$
$\Rightarrow B=5$
Putting $x-2=0\Rightarrow x=2$ in equation (2)
$\Rightarrow 2^2-5\times 2-1=0+0+C(2-1{)}^2$
$\Rightarrow 4-10-1=C$
$\Rightarrow C=-7$
Comparing the coefficients of $x^2$ in both the sides in equation (2) we get.
$\Rightarrow 1=A+C$
Putting the value of C from equation (1)
$\frac{x^2-5x-1}{(x-1{)}^2(x-2)}=\frac{8}{x-1}+\frac{5}{(x-1{)}^2}-\frac{7}{x-2}$.