Factorize the polynomial $x^3+y^3+z^3-3xyz$ I want to factorize the polynomial $x^3+y^3+z^3-3xyz$. Using Mathematica I find that it equals $(x+y+z)(x^2+y^2+z^2-xy-yz-zx)$. But how can I factorize it by hand?

$$\underbrace{x^3+y^3}+z^3-3xyz = \underbrace{(x+y)^3-3xy(x+y)}+z^3-3xyz$$
$$=\underbrace{(x+y)^3+z^3}-\underbrace{3xy(x+y)-3xyz} $$
$$=\underbrace{\{(x+y)+z\}}\{(x+y)^2-(x+y)z+z^2\}-3xy\underbrace{\{(x+y)+z\}} \left(\text{ using } a^3+b^3=(a+b)(a^2-ab+b^2)\text{ for the first two terms }\right)
=(x+y+z)\{(x+y)^2-(x+y)z+z^2-3xy\}$$
Now $\displaystyle (x+y)^2-(x+y)z+z^2-3xy=x^2+y^2+z^2-xy-yz-zx$
