If 6 x^3+7 x^2+6 x-3 (x-1)(x+3) (x^2+1 ) = A x-1 + B x+3 + C x+D x^2+1 and n=A+B+C+D and ⁵⁰ C_n= ⁵⁰ C_r , then r is equal to

Step-by-step solution

Given, 6 x^3+7 x^2+6 x-3 (x-1)(x+3) (x^2+1 ) = A x-1 + B x+3 + (C x+D) x^2+1 6 x^3+7 x^2+6 x-3 aligned & =A(x+3) (x^2+1 )+B(x-1) (x^2+1 ) & +C x(x-1)(x+3)+D(x-1)(x+3) aligned Put x=1, A=2 Put x=-3, B=3 Put x=0, D=2 Put x=-1, A=3, B=3, D=2 , we get C=1 n=A+B+C+D=2+3+1+2=8 ⁵⁰ C_n= ⁵⁰ C_r n+r=50r=52-n=50-8=42