Using the properties of determinants, show that:
`[[x, x^2, yz],[y, y^2, zx],[z, z^2, xy]]=(x-y)(y-z)(z-x)(xy+yz+zx)`

Explanation

`L.H.S.=|{:(x,x^(2),yz),(y,y^(2),zx),(z,z^(2),xy):}|=|{:(x-y,x^(2)-y^(2),yz-zx),(y-z,y^(2)-z^(2),zx-xy),(z,z^(2),xy):}|`
`(R_(1)toR_(1)-R_(2),R_(2)toR_(2)-R_(3))`
`=|{:(x-y,(x-y)(x+y),-z(x-y)),(y-z,(y-z)(y+z),-x(y-z)),(z,z^(2),xy):}|`
`=(x-y)(y-z)|{:(1,x+y,-z),(0,z-x,z-x),(0,-yz,xy+zx):}|`
`(R_(2)toT_(2)-R_(1),R_(3)toR_(3)-zR_(3))`
`=(x-y)(y-z)(z-x)|{:(1,x+y,-z),(0,z-x,z-x),(0,-yz,xy+zx):}|`
`=(x-y)(y-z)(z-x).1|{:(1,1),(-yz,xy-zx):}|`
(Expanding along `C_(1)`)
=(x-y)(y-z)(z-x)(xy+zx+yz)
=(x-y)(y-z)(z-x)(xy+yz+zx)