Solve this

$\Delta=\left|\begin{array}{ccc}0 & x y z & x-z \\ y-x & 0 & y-z \\ z-x & z-y & 0\end{array}\right|$

Expanding along $R_{1}$, we get

$=0(0-(y-x)(z-y))-(y-z)(0-(x-z)(z-y))+(z-x)(x y z(y-z)-0)$

$=0-(y-x)(-(x-z)(z-y))+(z-x)(x y z(y-z))$

$=(y-x)(x-z)(z-y)+(z-x) x y z(y-z)$

$=(y-z)(z-x)(y-x+x y z)$

Hence, $\left|\begin{array}{ccc}0 & x y z & x-z \\ y-x & 0 & y-z \\ z-x & z-y & 0\end{array}\right|=\underline{(y-z)(z-x)(y-x+x y z)}$