Show that `|(x,y,z),(x^(2),y^(2),z^(2)),(x^(3),y^(3),z^(3))|=xyz (x-y) (y-z) (z-x)`

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

Solve that |(y+z,x,x^(2)),(z+x,y,y^(2)),(x+y,z,z^(2))|=(x+y+z)(x-y)(y-z)(z-x)

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

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

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

Let |{:(y^(5)z^(6)(z^(3)-y^(3)),,x^(4)z^(6)(x^(3)-z^(3)),,x^(4)y^(5)(y^(3)-x^(3))),(y^(2)z^(3)(y^(6)-z^(6)),,xz^(3)(z^(6)-x^(6)) ,,xy^(2)(x^(6)-y^(6))),(y^(2)^(3)(z^(3)-y^(3)),,xz^(3)(x^(3)-z^(3)),,xy^(2)(y^(3)-x^(3))):}| " and " Delta_(2)= |{:(x,,y^(2),,z^(3)),(x^(4),,y^(5) ,,z^(6)),(x^(7),,y^(8),,z^(9)):}| .Then Delta_(1)Delta_(2) is equal to

it x_(1)^(2) +2y_(1)^(2)+3z_(1)^(2)=x_(2)^(2)+2y_(2)^(2)+3z_(2)^(2)=x_(3)^(2)+2y_(3)^(2)+3z_(3)^(2)=2 " and " x_(2)x_(3) +2y_(2)y_(3)+3z_(2)z_(3)=x_(3)x_(1)+2y_(3)y_(1)+3z_(3)z_(1)=x_(1)x_(2)+2y_(1)y_(2)+3z_(1)z_(2)=1 Then find the value of |{:(x_(1),,y_(1),,z_(1)),(x_(2),,y_(2),,z_(2)),(x_(3),,y_(3),,z_(3)):}|

By using 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)

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

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