<s>[INST] 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?
 [/INST] Consider the polynomial 
$$(\lambda - x)(\lambda - y)(\lambda - z) = \lambda^3 - a\lambda^2+b\lambda-c\tag{*1}$$
We know
$$\begin{cases}a = x + y +z\\ b = xy + yz + xz \\ c = x y z\end{cases}$$ 
Substitute  $x, y, z$ for $\lambda$ in $(*1)$ and sum, we get
$$x^3 + y^3 + z^3 - a(x^2+y^2+z^2) + b(x+y+z) - 3c = 0$$
This is equivalent to
$$\begin{align}
  x^3+y^3+z^3 - 3xyz
= & x^3+y^3+z^3 - 3c\\
= & a(x^2+y^2+z^2) - b(x+y+z)\\
= & (x+y+z)(x^2+y^2+z^2 -xy - yz -zx)
\end{align}$$
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