aligned & If x^3 (2 x-1)(x-1)^2 =A+ B 2 x-1 + C x-1 & + D (x-1)^2 , then 2 A-3 B+4 C+5 D= aligned

Step-by-step solution

We have, x^3 (2 x-1)(x-1)^2 = x^3 2 x^3-5 x^2+4 x-1 Now, consider aligned & 1 2 ( 5 x^2-4 x+1 2 x^3-5 x^2+4 x-1 ) & = 1 2 ( B 2 x-1 + C x-1 + D (x-1)^2 ) & 1 2 ( 5 x^2-4 x+1 2 x^3-5 x^2+4 x-1 ) & = 1 2 ( B(x-1)^2+C(x-1)(2 x-1)+D(2 x-1) (2 x-1)(x-1)^2 ) & 5 x^2-4 x+1 & =B(x-1)^2+C(x-1)(2 x-1)+D(2 x-1) aligned Put x=1 , we get D=2 Put x=1 / 2 , we get B=+1 Now put x=0 , we get aligned 1 & =B+C-D 1 & =+1+C-2 C=2 aligned Now, 1 2 ( 5 x^2-4 x+1 (2 x-1)(x-1)^2 ) gathered = 1 2 ( +1 2 x-1 + 2 x-1 + 2 (x-1)^2 ) = + 1 2 2 x