$\begin{array}{l}x^3 + y^3 + z^3 - 3xyz\\ = (x+y)^3 - 3xy(x+y) + z^3 - 3xyz\\ = (x+y+z)[(x+y)^2 - (x+y).z + z^2] - 3xy(x+y + z)\\ = (x+y+z)(x^2 + 2xy + y^2 - xz - yz + z^2 - 3xy)\\ = (x+y+z)(x^2 + y^2 + z^2 - xy - yz - zx)\\ = \dfrac{1}{2}(x+y+z)(2x^2 + 2y^2 + 2z^2 - 2xy - 2yz - 2zx)\\ = \dfrac{1}{2}(x+y+z)[(x-y)^2 + (y-z)^2 + (z-x)^2]\qquad (đpcm) \end{array}$