aligned & If 4 x^2+5 x^4+7 (x^2+1 ) (x^4+x^2+1 ) = A x+B x^2+1 & + C x^3+D x^2+E x+F x^4+x^2+1 , then & B+2(D+F+E)-C A= aligned

Step-by-step solution

We have, aligned & 4 x^2+5 x^4+7 (x^2+1 ) (x^4+x^2+1 ) = A x+B x^2+1 + C x^3+D x^2+E x+F x^4+x^2+1 & 4 x^2+5 x^4+7=(A x+B) (x^4+x^2+1 ) & + (C x^3+D x^2+E x+F ) (x^2+1 ) & 5 x^4+4 x^2+7=A x^5+A x^3+A x+B x^4+B x^2 & +B+C x^5+C x^3+D x^4+D x^2+E x^3+E x+F x^2+F & 5 x^4+4 x^2+7=(A+C) x^5+(B+D) x^4 & +(A+C+E) x^3+(B+D+F) x^2+(A+E) x+(B+F) aligned On comparing, we get aligned & A+C=0 & B+D=5 aligned array r A+C+E=0 B+D+F=4 A+E=0 B+F=7 array On solving, we get gathered A=C=E=0, B=8 . D=-3, F=-1 B+2(D+F+E)-C A=8+2(-3-1+0