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

Do you know, $$x^3-y^3=(x-y)(x^2+xy+y^2)$$ and not $$x^3-y^3\ne(x-y)(x^2+y^2)$$ in general?
