Question

Solve:$det\left|\begin{array}{ccc}x& x^2& yz\\ y& y^2& zx\\ z& z^2& xy\end{array}\right|$

Answer 1

$det\left|\begin{array}{ccc}x& x^2& yz\\ y& y^2& zx\\ z& z^2& xy\end{array}\right|$

$=xdet\left(\begin{array}{cc}{y}^2& zx\\ z^2& xy\end{array}\right)-x^{2}det\left(\begin{array}{cc}y& zx\\ z& xy\end{array}\right)+yzdet\left(\begin{array}{cc}y& y^2\\ z& z^2\end{array}\right)$

$det\left(\begin{array}{cc}{y}^2& zx\\ z^2& xy\end{array}\right)=x{y}^3-x{z}^3$

$det\left(\begin{array}{cc}y& zx\\ z& xy\end{array}\right)=x{y}^2-x{z}^2$

$det\left(\begin{array}{cc}y& y^2\\ z& z^2\end{array}\right)=y{z}^2-y^{2}z$

$=x(x{y}^3-x{z}^3)-x^2(x{y}^2-x{z}^2)+yz(y{z}^2-y^{2}z)$

$=\left(x-y\right)\left(y-z\right)\left(z-x\right)\left(xy+yz+zx\right)$