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

Show 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)

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

The value of |(1,1,1),(2x,2y,2z),(3x,3y,3z)| is ____

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)]

If sin^(-1)x+sin^(-1)y+sin^(-1)z=pi , then prove that x^(4)+y^(4)+z^(4)+4x^(2)y^(2)z^(2)=2(x^(2)y^(2)+y^(2)z^(2)+z^(2)x^(2))

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)