$\,\,\,\,\,yz\left( y-z \right)-xz\left( x-z \right)+xy\left( x-y \right)$
$={{y}^{2}}z\,\,-\,\,y{{z}^{2}}\,\,-{{x}^{2}}z\,\,+\,\,x{{z}^{2}}\,\,+\,\,xy\left( x-y \right)$
$=\left( {{y}^{2}}z\,\,-\,\,{{x}^{2}}z \right)\,\,+\,\,\left( x{{z}^{2}}\,\,-\,\,y{{z}^{2}} \right)\,\,+\,\,xy\left( x-y \right)$
$=z\left( {{y}^{2}}-{{x}^{2}} \right)\,\,+\,\,{{z}^{2}}\left( x-y \right)\,\,+\,\,xy\left( x-y \right)$
$=-z\left( {{x}^{2}}-{{y}^{2}} \right)\,\,+\,\,{{z}^{2}}\left( x-y \right)\,\,+\,\,xy\left( x-y \right)$
$=-z\left( x-y \right)\left( x+y \right)\,\,+\,\,{{z}^{2}}\left( x-y \right)\,\,+\,\,xy\left( x-y \right)$
$=\left( x-y \right)\,\,\left[ -z\left( x+y \right)\,\,+\,\,{{z}^{2}}\,\,+\,\,xy \right]$
$=\left( x-y \right)\left( -zx\,\,-\,\,zy\,\,+\,\,{{z}^{2}}\,\,+\,\,xy \right)$
$=\left( x-y \right)\left[ \,\left( -zx\,\,+\,\,{{z}^{2}} \right)\,\,+\,\,\left( xy-zy \right)\, \right]$
$=\left( x-y \right)\left[ \,-z\left( x\,\,-\,\,z \right)\,\,+\,\,y\left( x\,\,-\,\,z \right) \right]$
$=\left( x-y \right)\left( x-z \right)\left( y-z \right)$
