Question

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

Answer 1

Answer:

1/2(x+y+z)(x-y)^2+(y-z)^2+(z-x)^2

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

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

Hence,we verify that

(x^3+y^3+z^3-3xyz)=1/2(x+y+z){(x-y)^2+(y-z)^2+(z-x)^2}