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
$$ (a-b)(b-c)(c-a)(a+b+c) $$

Expand this
$$\begin{vmatrix}
1 &1 &1 \\
a &b &c \\
a^3 &b^3 &c^3 \\
\end{vmatrix}$$
to an expanded polynomial expression. Now expand this $$ (a-b)(b-c)(c-a)(a+b+c) $$
to an expanded polynomial expression.
Now check that they're both equal.
