\left |\begin{array}{ccc}
1 & x & x^2\\
1 & y & y^2\\
1 & z & z^2
\end{array}\right | =  \left |\begin{array}{ccc}
1 & x & 0\\
1 & y & y^2-yx\\
1 & z & z^2-zx
\end{array}\right | = \left |\begin{array}{ccc}
1 & 0 & 0\\
1 & y-x & y(y-x)\\
1 & z-x & z(z-x)
\end{array}\right | = \left |\begin{array}{cc}
y-x & y(y-x)\\
z-x & z(z-x)
\end{array}\right | = (y-x)(z-x)\left |\begin{array}{cc}
1 & y\\
1 & z
\end{array}\right | = (y-x)(z-x)(z-y)