aligned & If x^4 (x-a)(x-b)(x-c) =P(x)+ A x-a + B x-b & + C x-c , then P(0)+A(a-b)(a-c)= aligned

Step-by-step solution

If is given that array r x^4 (x-a)(x-b)(x-c) =P(x)+ A x-a + B x-b + C x-c x^4=(x-a)(x-b)(x-c) P(x)+A(x-b)(x-c) +B(x-c)(x-a)+C((x-a)(x-b) array At x=0, a b c P(0)=b c A+c a B+a b C At x=a, A= a^4 (a-b)(a-c) , similarly At x=b, B= b^4 (b-c)(b-a) and So, aligned & P(0)= a^3 (a-b)(a-c) + b^3 (b-c)(b-a) + c^3 (c-a)(c-b) & = a^3(c-b)+b^3(a-c)+c^3(b-a) (a-b)(b-c)(c-a) & = a^3(c-b)+b c (c^2-b^2 )+a (b^3-c^3 ) (a-b)(b-c)(c-a) & = a^3+b c(c+b)-a (b^2+c^2+b c ) (a-b(a-c) & = a (a^2-b^2 )-c^2(a-b)-b c(a-b) (a-b)(a-c) & = (a^2+