What is the value of this determinant? [2]

Question

If $xyz = -1$, what is the value of the determinant $$\begin{vmatrix} x & x^2 & x^3+1\\ y & y^2 & y^3+1 \\ z & z^2 & z^3+1 \end{vmatrix}$$

I need a solution without putting random values of x y and z. PS don't tell to open the determinant

Answer 1

$\det\begin{pmatrix} x & x^2 & x^3+1 \\ y & y^2& y^3+1 \\ z & z^2 &z^3+1\end{pmatrix}$

$=\det\begin{pmatrix} x & x^2 & x^3 \\ y & y^2& y^3 \\ z & z^2 &z^3 \end{pmatrix}+\det\begin{pmatrix} x & x^2 & 1 \\ y & y^2& 1 \\ z & z^2 & 1\end{pmatrix}$

$=xyz\cdot \det\begin{pmatrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z &z^3 \end{pmatrix}+\det\begin{pmatrix} x & x^2 & 1 \\ y & y^2& 1 \\ z & z^2 & 1\end{pmatrix}$

Now $\det\begin{pmatrix} x & x^2 & 1 \\ y & y^2& 1 \\ z & z^2 & 1\end{pmatrix}=-\det\begin{pmatrix} x &1& x^2 \\ y &1 & y^2 \\ z &1 & z^2 \end{pmatrix}=\det\begin{pmatrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z &z^2 \end{pmatrix}$