<s>[INST] Factorise the determinant $\det\Bigl(\begin{smallmatrix} a^3+a^2 & a & 1 \\ b^3+b^2 & b & 1 \\ c^3+c^2 & c &1\end{smallmatrix}\Bigr)$ Factorise the determinant $\det\begin{pmatrix} a^3+a^2 & a & 1 \\ b^3+b^2 & b & 1 \\ c^3+c^2 & c &1\end{pmatrix}$.
My textbook only provides two simple examples.
Really have no idea how to do this type of questions..
 [/INST] $\det\begin{pmatrix} a^3+a^2 & a & 1 \\ b^3+b^2 & b & 1 \\ c^3+c^2 & c &1\end{pmatrix}$
$=\det\begin{pmatrix} a^3+a^2 & a & 1 \\ b^3+b^2-(a^3+a^2) & b-a & 1-1 \\ c^3+c^2-(a^3+a^2) & c-a &1-1\end{pmatrix} $   (applying $R_2'=R_2-R_1\ and\  R_3'=R_3-R_1$)
$=(b-a)(c-a)
\det\begin{pmatrix} a^3+a^2 & a & 1 \\ b^2+a^2+ab+b+a & 1 & 0 \\ c^2+a^2+ca+c+a & 1 & 0\end{pmatrix}$
$=(b-a)(c-a)\cdot1\cdot
\det\begin{pmatrix}  b^2+a^2+ab+b+a & 1 & \\ c^2+a^2+ca+c+a & 1\end{pmatrix}$
$=(b-a)(c-a)(b-c)(a+b+c+1)$
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