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

We know that, $$x^3 + y^3 + z^3 - 3xyz = (x + y + z)[x^2 + y^2 + z^2 - xy - yz - zx]$$
$$= 1/2(x + y + z)[2x^2 + 2y^2 + 2z^2 - 2xy - 2yz - 2zx]$$
$$= 1/2(x + y + z)[x^2 + x^2 + y^2 + y^2 + z^2 + z^2 - 2xy - 2yz - 2zx]$$
$$= 1/2(x + y + z)[x^2 + y^2 - 2xy + y^2 + z^2 - 2yz + z^2 + x^2 - 2zx]$$
$$= 1/2(x + y + z)[(x - y)^2 + (y - z)^2 + (z - x)^2]$$