<s>[INST] Factorising a 3 x 3 determinant - What Am I doing Wrong? $$\begin{vmatrix}
1 & a & a^3 \\
1 & b & b^3 \\
1 & c & c^3 \\
\end{vmatrix}$$
subtracting the top row from the middle and bottom rows
$$ = 
\begin{vmatrix}
1 & a & a^3 \\
0 & (b - a) & (b^3 - a^3)\\
0 & (c - a) & (c^3 - a^3)\\
\end{vmatrix}
$$
expanding the determinant
$$
= \begin{vmatrix}
(b - a) & (b^3 - a^3)\\ 
(c - a) & (c^3 - a^3)\\
\end{vmatrix}
$$
pulling out factors
$$
= (b-a)(c-a)
\begin{vmatrix}
1 & (b^2 + a^2)\\ 
1 & (c^2 + a^2)\\
\end{vmatrix}
$$
and evaluating gives
$$
= (b-a)(c-a)(c^2 - b^2)
$$
yet the answer in the book gives
$$(a+b+c)(a-b)(b-c)(c-a)$$
I am self studying so thanks in advance
 [/INST] But $b^3-a^3 = (b-a)(b^2+ab+a^2)$ and $c^3-a^3 = (c-a)(c^2+ac+a^2)$. Hence
$$(c^2+ac+a^2) - (b^2+ab+a^2) = c^2+ac- b^2-ab = (c-b)(a+b+c).$$
</s>