<s>[INST] Who came up with the identity $a^3+b^3+c^3-3abc=(a+b+c)\left[a^2+b^2+c^2-ab-bc-ca\right]$ Though we can prove this it is not something that comes up intutively.
Our ancestors must have been interested in factorising $a^3+b^3+c^3$ but why find it for $a^3+b^3+c^3-3abc$ ?
 [/INST] I don't know the history of this identity, but here is a derivation based on the properties of determinants.
$$\begin{vmatrix}
a &c &b\\
b &a &c\\
c &b &a
\end{vmatrix} =
\begin{vmatrix}
a+b+c &a+b+c &a+b+c\\
b &a &c\\
c &b &a
\end{vmatrix} = (a+b+c) \begin{vmatrix}
1 &1 & 1\\
b &a &c\\
c &b &a
\end{vmatrix} $$
Expanding the determinant on the left, we have
$$a^3+b^3+c^3-3abc$$
and expanding the determinant on the right we have
$$(a+b+c)(a^2+b^2+c^2-ab-ac-bc)$$
so these two expressions are equal.
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