If ( 3 (x-1) (x^2+x+1 ) = 1 x-1 ) ( aligned & - x+2 x^2+x+1 =f₁(x)-f₂(x) and x+1 (x-1)^2 (x^2+x+1 ) =A f₁(x)+ (B+ D x-1 ) & f₂(x)+ C (x-1)^2 , A+B+C+D= aligned )

Step-by-step solution

It is given that ( aligned & 3 (x-1) (x^2+x+1 ) = 1 x-1 - x+2 x^2+x+1 & =f₁(x)-f₂(x) & and x+1 (x-1)^2 (x^2+x+1 ) =A f₁(x)+ (B+ D x-1 ) & f₂(x)+ C (x-1)^2 aligned ) From above informations, it is necessary, that ( aligned f₁(x) & = 1 x-1 and f₂(x) & = x+2 x^2+x+1 aligned ) So, ( array r x+1 (x-1)^2 (x^2+x+1 ) = A x-1 + (B x-B+D)(x+2) (x^2+x+1 )(x-1) + C (x-1)^2 array ) ( aligned x+1 & =A(x-1) (x^2+x+1 )+B(x-1) & (x+2)(x-1)+D(x+2)(x-1)+C (x^2+x+1 ) aligned ) On comparing the coefficient, we get coefficient of (x^3=0