Factorise a matrix using the factor theorem Can someone check this please?
 $$
\begin{vmatrix}
x&y&z\\
x^2&y^2&z^2\\
x^3&y^3&z^3\\
\end{vmatrix}$$
$$C_2=C_2-C_1\implies\quad
\begin{vmatrix}
x&y-x&z\\
x^2&y^2-x^2&z^2\\
x^3&y^3-x^3&z^3\\
\end{vmatrix}$$
$$(y-x)
\begin{vmatrix}
x&1&z\\
x^2&y+x&z^2\\
x^3&y^2+xy+x^2&z^3\\
\end{vmatrix}$$
$$(y-x)(z-x)
\begin{vmatrix}
x&1&1\\
x^2&y+x&z+x\\
x^3&y^2+xy+x^2&z^2+xz+x^2\\
\end{vmatrix}$$
$$R_2=R_2-xR_1\implies\quad
(y-x)(z-x)
\begin{vmatrix}
x&1&1\\
0&y&z\\
x^3&y^2+xy+x^2&z^2+xz+x^2\\
\end{vmatrix}$$
$$R_3=R_3-x^2R_1\implies\quad
(y-x)(z-x)
\begin{vmatrix}
x&1&1\\
0&y&z\\
0&y^2+xy&z^2+xz\\
\end{vmatrix}$$
factor $x$$$\implies\quad
x(y-x)(z-x)
\begin{vmatrix}
1&1&1\\
0&y&z\\
0&y^2+xy&z^2+xz\\
\end{vmatrix}$$
$$\implies\quad x(y-x)(z-x)(yz^2-zy^2)$$
$$\implies\quad xyz(y-x)(z-x)(z-y)$$
Also I'd like practical tips on using the factor theorem for these types of questions. My understanding is that the determinant is $f(x,y,z)$ so if we hold $y$ and $z$ constant we could apply it somehow to $f(x)$ alone. I'm not that great spotting difference of squares etc and want a more fail safe alternative. Thanks in advance.

What you did is correct. But there is an easier way. Remember that for polynomial $p(x)$, if $p(a)=0$ then $(x-a)$ is a factor of $p(x)$.
Denote the determinant by $\Delta$. It is obviously a polynomial in $x,\ y$ and $z$. Now, note that:


*

*$x=0\implies \Delta = 0$, so $x$ is a factor of $\Delta$. Same for $y = 0$ and $z=0$.

*$x=y\implies \Delta = 0$, so $(x-y)$ is a factor of $\Delta$. Similarly for $y=z$ and $z=x$


Finally note that $\Delta$ is degree $6$ polynomial. So it cannot have more than $6$ linear factors, and we have listed all of them above. Clearly
$$\Delta=Cxyz(x-y)(y-z)(z-x)$$
where $C$ is some constant. Taking some values (eg. $x=1,\ y=2,\ z=3$), we get $C=1$.
