Question

Verify that
x³+y³+z³-3xyz = 1/2(x+y+z)[ (x-y)² + (y-z)² + (z-x) ² ]​

Answer 1

x^3+y^3+z^3-3xyz=1/2(x+y+z)[(x-y)^2+(y-z)^2+(z-x)^2)]

RHS= 1/2(x+y+z)(x^2-2xy+y^2+y^2-2xy+z^2+z^2-2zx+x^2)

=1/2(x+y+z)(2x^2+2y^2+2z^2-2xy-2yz-2zx)

=1/2(x+y+z)×2(x^2+y^2+z^2-xy-yz-zx)

=( x+y+z) (x^2+y^2+z^2-xy-yz-zx).

= x^3+y^3+z^3-3xyz

therefore, LHS=RHS.