Question

Prove that (x + y) cube - (x - y) cube - 6y (x square - y square) = 8y cube

Answer 1

lhs = (x+y)^3 - (x-y)^3 -6y(x^2 -y^2)

= x^3 +3x^2y+3xy^2 +y^3 - (x^3-3x^2y+3xy^2-y^3)-6x^2y+6y^3
=x^3 +3x^2y+3xy^2+y^3-x^3+3x^2y-3xy^2 +y^3 -6x^2y+6y^3

= x^3 -x^3 +6x^2y-6x^2y +3x^2y-3x^2y +2y^3+6y^3
= 8y^3
=rhs