verify that x^3+y^3+z^3-3xyz=1/2(x+y+z){(x-y)^2+(y-z)^2+(z-x)^2} . Please solve this with explanation
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L.H.S=x^3+y^3+z^3-3xyz
=(x+y+z)(x^2+y^2+z^2-xy-yx+-zx) =1/2(x+y+z) (2x^2+2y^2+2z^2-2xy-2yz-2zx)
[by multiplying and dividing by 2]
=1/2(x+y+z) {(x-y)^2+(y-z)^2+(z-x)^2} = R.H.S
(proved)
=(x+y+z)(x^2+y^2+z^2-xy-yx+-zx) =1/2(x+y+z) (2x^2+2y^2+2z^2-2xy-2yz-2zx)
[by multiplying and dividing by 2]
=1/2(x+y+z) {(x-y)^2+(y-z)^2+(z-x)^2} = R.H.S
(proved)
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