Without expanding show that : `|(x-y,y-z,z-x),(y-z,z-x,x-y),(z-x,x-y,y-z)|=0`

Using properties of determinant show that : |(y+z,x,x),(y,z+x,y),(z,z,x+y)|=4xyz

Using properties of determinant show that : |((x+y)^2,zx,zy),(zx,(z+y)^2,xy),(zy,xy,(z+x)^2)|=2xyz(x+y+z)^3

Find the value of : |(1+x,y,z),(x,1+y,z),(x,y,1+z)|

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

Factorise : |(1,1,1),(x,y,z),(x^3,y^3,z^3)|

Factorise : |(1,1,1),(x,y,z),(x^2,y^2,z^2)|

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

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

Prove that (3x-2y)^3+(y+2z)^3+(y-2z-3x)^3=3(3x-2y)(y+2z)(y-2z-3x)

Factorise (x-y)^3+(y-z)^3+(z-x)^3