1.

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

Answer» `R.H.S. = 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^2-2zx)`
`=1/2(x+y+z)(2x^2+2y^2+2z^2-2xy-2yz-2zx)`
`=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)`
We have the formula,
`x^3+y^3+z^3 -3xyz =(x+y+z)(x^2+y^2+z^2-xy-yz-zx)`
Thus, `L.H.S. = R.H.S.`


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