Show the determinant of the matrix is $(a − b)(b − c)(c − a)$

Question

Show that the determinant$$ \begin{vmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \\ \end{vmatrix} = (a − b)(b − c)(c − a)$$

Using the Cofactor expansion I've got $a((b-c)(b+c) + a(c-b)) + b(c(c-b))$ but I dont know what else to do.

Answer 1

Substract first row from second and third ones and get:

$$\begin{vmatrix} 1&a&a^2\\ 0&b-a&b^2-a^2\\ 0&c-a&c^2-a^2\end{vmatrix}=(b-a)(c-a)\begin{vmatrix} 1&a&a^2\\ 0&1&b+a\\ 0&1&c+a\end{vmatrix}=$$

$$=(b-a)(c-a)\begin{vmatrix} 1&b+a\\ 1&c+a\end{vmatrix}=(b-a)(c-a)\left(\begin{vmatrix} 1&b\\ 1&c\end{vmatrix}+\overbrace{\begin{vmatrix} 1&a\\ 1&a\end{vmatrix}}^{=0}\right)=$$

$$=(b-a)(c-a)(c-b)$$