If 2 x^3+x^2-5 x^4-25 = A x+B x^2-5 + C x+1 x^2+5 , then (A, B, C) equals to

Step-by-step solution

aligned & Given, 2 x^3+x^2-5 x^4-25 = A x+B x^2-5 + C x+1 x^2+5 & 2 x^3+x^2-5=(A x+B) (2 x^2+5 ) & +(C x+1) (x^2-5 ) & 2 x^3+x^2-5=A x^3+5 A x+B x^2+5 B & +C x^3-5 C x+x^2-5 & 2 x^3+x^2-5=x^3(A+C)+x^2(B+1) & +x(5 A-5 C)+5 B-5 aligned On equating the coefficients of x^3, x^2, x and constant, we get aligned & A+C=2, B+1=1,5 A-5 C=0 & and & 5 B-5=-5 & B+1=1 B=0 & 5 A-5 C=0 A=C & A+C=2 C+C=2 & 2 C=2 C=1 aligned Now, and aligned & & A & =C=1 & & (A, B, C) & =(1,0,1) aligned