Question

resolve into partial fractions x/(x-1)(x-2)

Answer 1

Please find the attached image of solution.

Answer 2

$\fontsize{18}{10}{\textup{\textbf{The required partial fraction form is}}}$

$\dfrac{x}{(x-1)(x-2)}=-\dfrac{1}{x-1}+\dfrac{2}{x-2}.$

Step-by-step explanation:  Let us consider that

$\dfrac{x}{(x-1)(x-2)}=\dfrac{A}{x-1}+\dfrac{B}{x-2}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)$

Multiplying both sides of equation (i) by $(x-1)(x-2),$ we have

$x=A(x-2)+B(x-1)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)$

If x = 1, the we get from equation (ii) that

$1=A(1-2)\\\\\Rightarrow A=-1.$

If x = 2, the we get from equation (ii) that

$2=B(2-1)\\\\\Rightarrow B=2.$

Thus, the required partial fraction form is

$\dfrac{x}{(x-1)(x-2)}=-\dfrac{1}{x-1}+\dfrac{2}{x-2}.$