Show that $x$, $y$, and $z$ are not distinct if $x^2(z-y) + y^2(x-z) + z^2(y-x) = 0$.

Question

Suppose that $x^2(z-y) + y^2(x-z) + z^2(y-x) = 0$.

How can I show that $x$, $y$, and $z$ are not all distinct, that is, either $x=y$, $y=z$, or $x=z$?

Answer 1

$x^2(z-y)+y^2(x-z)+z^2(y-x)=x^2(z-y)+x(y^2-z^2)+z^2y-y^2z=(z-y)(x^2-x(y+z)+zy)=(z-y)(x-y)(x-z)$.