<s>[INST] Prove that the determinant is $(a-b)(b-c)(c-a)(a+b+c)$ I have the determinant : 
\begin{vmatrix}
1 &1 &1 \\
a &b &c \\
a^3 &b^3 &c^3 \\
\end{vmatrix}
How do I prove that this determinant is equal to
$$ (a-b)(b-c)(c-a)(a+b+c) $$
 [/INST] HINT:
Use $$C_1'=C_1-C_3$$ and $$C_2'=C_2-C_3$$
where $C_r$ is the $r$th column, $C_r'$ is the resultant $r$th column
See also: Factorise the determinant $\det\Bigl(\begin{smallmatrix} a^3+a^2 & a & 1 \\ b^3+b^2 & b & 1 \\ c^3+c^2 & c &1\end{smallmatrix}\Bigr)$
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