Question

Prove that $\left|\begin{array}{ccc}1& x& x^3\\ 1& y& y^3\\ 1& z& z^3\end{array}\right|=(x+y+z)(x-y)(y-z)(z-x)$

Answer 1

$\left|\begin{array}{ccc}1& x& x^3\\ 1& y& y^3\\ 1& z& z^3\end{array}\right|=1(y{z}^3-z{y}^3)-x\left({z^3-y}^3\right)+x^3(z-y)$

$\Rightarrow yz(z-y)(z+y)-x(z-y)(z^2+zy+y^2)+x^3(z-y)$

$\Rightarrow (z-y)(x^3-{xz}^2-xzy-x{y}^2+y{z}^2+z{y}^2)$

$\Rightarrow (z-y)(z-x)(yz-x(x+z)+y^2)$

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