Question

Докажите тождество)
(a+b)^2(a-b)-2ab(b-a)-6ab(a-b)=(a-b)^3

Answer 1

(a+b)^2 × a - (a+b)^2 × b - 2ab^2 + 2a^2b - 6a^2b + 6ab^2 = (a^2 + 2ab + b^2) × a - (a^2 + 2ab + b^2) × b - 2ab^2 +2a^2b - 6a^2b + 6ab^2 = a^3 + 2a^2b + ab^2 - (a^2 + 2ab + b^2) × b - 2ab^2 + 2a^2b - 6a^2b + 6ab^2 = a^3 + 2a^2b + 5ab^2 + (-a^2 - 2ab - b^2) ×b = a^3 + 2a^2b + 5ab^2 - a^2b - 2ab^2 - b^3 = a^3 - 3a^2b + 3ab^2 - b^3 = (a - b)^3