Solution:

$\begin{array}{l}
\left|\begin{array}{ccc}
\mathrm{x} & \mathrm{y} & \mathrm{z} \\
\mathrm{x}^{2} & \mathrm{y}^{2} & \mathrm{z}^{2} \\
\mathrm{x}^{3} & \mathrm{y}^{3} & \mathrm{z}^{3}
\end{array}\right| \\
=\mathrm{xyz}\left|\begin{array}{ccc}
1 & 1 & 1 \\
\mathrm{x} & \mathrm{y} & \mathrm{z} \\
\mathrm{x}^{2} & \mathrm{y}^{2} & \mathrm{z}^{2}
\end{array}\right|
\end{array}$
$\begin{array}{l}
=x y z\left|\begin{array}{ccc}
0 & 0 & 1 \\
x-y & y-z & z \\
x^{2}-y^{2} & y^{2}-z^{2} & z^{2}
\end{array}\right|\left[C_{1}^{\prime}=C_{1}-C_{2} \& C_{2}^{\prime}=C_{2}-C_{3}\right] \\
=x y z\left|\begin{array}{ccc}
x-y & y-z & z \\
(x+y)(x-y) & (y+z)(y-z) & z^{2}
\end{array}\right| \\
=x y z(x-y)(y-z)\left|\begin{array}{ccc}
1 & 1 & z \\
x+y & y+z & z^{2}
\end{array}\right|\left[C_{1}^{\prime}=C_{1} /(x-y) \& C_{2}^{\prime}=C_{2} /(y-z)\right] \\
=x y z(x-y)(y-z)(0+0+y+z-x-y)[\text { expansion by first row }] \\
=x y z(x-y)(y-z)(z-x)
\end{array}$