Prove that `|[1,x,x^2-yz],[1,y,y^2-zx],[1,z,z^2-xy]|= 0`

Prove that: |[x,x^2,yz],[y,y^2,zx],[z,z^2,xy]|=(x-y)(y-z)(z-x)(xy+yz+zx)

Using the properties of determinant, show that : |[1,x+y,x^2+y^2],[1,y+z,y^2+z^2],[1,z+x,z^2+x^2]| = (x-y)(y-z)(z-x)

Prove that |(1,x,x^2),(1,y,y^2),(1,z,z^2)| = (x-y)(y-z)(z-x)

Using Cofactors of elements of third column, evaluate triangle - |[1,x,yz],[1,y,zx],[1,z,xy]|

Prove that |[[x,y,z],[x^2,y^2,z^2],[x^3,y^3,z^3]]|= xyz (x-y)(y-z)(z-x)

Prove that: {:|(1,x,x^3),(1,y,y^3),(1,z,z^3)| = (x-y)(y-z)(z-x)(x+y+z)

If xneynez and |[x,x^3,x^4-1],[y,y^3,y^4-1],[z,z^3,z^4-1]|=0 , then prove that : xyz(xy+yz+zx)=x+y+z

Using properties of determinants, prove that: |[x,x^2,1+px^3],[y,y^2,1+py^3],[z,z^2,1+pz^3]| = (1+pxyz)(x-y)(y-z)(z-x)

Using properties of determinants , prove that |(x^2+1,xy,zx),(xy,y^2+1,yz),(zx,yz,z^2+1)|=1+x^2+y^2+z^2