Question

Factorise
$x^2(y-z)+y^2(z-x)+z^2(x-y)$

Answer 1

Given : $x^2(y-z)+y^2(z-x)+z^2(x-y)$

Expanding the given expression, we have

$=x^{2}y-x^{2}z+y^{2}z-y^{2}x+z^{2}x-z^{2}y$

Rearranging the above equation

$=x^{2}y-x^{2}z-y^{2}x+z^{2}x+y^{2}z-z^{2}y$

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

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

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

Taking $(y-z)$ common

$=(y-z)[x^2-x(y+z)+yz]$

$=(y-z)[x^2-xy-xz+yz]$

$=(y-z)\left[x\right(x-y)-z(x-y\left)\right]$

Taking $(x-y)$ common

$=(y-z)\left[\right(x-z\left)\right(x-y\left)\right]$

$=(x-y)(y-z)(x-z)$