Solve in partial fraction

Dear Student,

\frac{1}{{(x - 1)}^{2} (x - 2)} = \frac{A x + B}{{(x - 1)}^{2}} + \frac{C}{x - 2} 1 = (A x + B) (x - 2) + C {(x - 1)}^{2} 1 = A x^{2} + (B - 2 A) x - 2 B + C (x^{2} - 2 x + 1) 1 = A x^{2} + (B - 2 A) x - 2 B + C x^{2} - 2 x C + C 0 x^{2} + 0 x + 1 = (A + C) x^{2} + (B - 2 A - 2 C) x + (C - 2 B) c o m p a i r i n g c o e f f i c i e n t s, w e g e t, (A + C) = 0 \Rightarrow A = - C . . . . . . . . . . . . e q (1) (B - 2 A - 2 C) = 0 \Rightarrow 2 B - 4 A - 4 C = 0 . . . . . . . . . e q (2) (C - 2 B) = 1 \Rightarrow 2 B = C - 1 . . . . . . . . . . . . . . . e q (3) S u b s t i t u t i n g t h e v a l u e o f A a n d 2 B f r o m e q (1) a n d (3) i n e q (2), w e g e t, C - 1 - 4 (- C) - 4 C = 0 C - 1 + 4 C - 4 C = 0 C = 1 S u b s t i t u t i n g t h e v a l u e o f C i n e q (1) a n d (3), w e g e t, A = - 1 a n d B = 0 P u t t i n g t h e v a l u e s i n i n i t i a l e q u a t i o n w e g e t, \frac{1}{{(x - 1)}^{2} (x - 2)} = \frac{- x}{{(x - 1)}^{2}} + \frac{1}{x - 2}

Regards,