Determinant form of `(x-y)(y-z)(z-x)(x+y+z)`

Determinant , form (x-y)(y-z)(z-x)(xy+yz+zx), of

Determinant (x-y)(y-z)(z-x)(x+y+z)

Simplify: x(x-y)+y(y-z)+z(z-x)

The simplified form of x(y-z)(y+z)+y(z-x)(z+x)+z(x-y)(x+y) is

Area of the triangle formed by (x,y+z),(y,z+x),(z,x+y) is

The repeated factor of the determinant |(y +z,x,y),(z +x,z,x),(x +y,y,z)| , is

The repeated factor of the determinant |(y +z,x,y),(z +x,z,x),(x +y,y,z)| , is

By using properties of determinants. Show that: |[x,x^2,y z],[ y, y^2,z x],[ z, z^2,x y]|=(x-y)(y-z)(z-x)(x y+y z+z x)

What is the determinant of the matrix ({:(x,y,y+z),(z,x,z+x),(y,z,x+y):}) ?

Prove the identities: |[z, x, y],[ z^2,x^2,y^2],[z^4,x^4,y^4]|=|[x, y, z],[ x^2,y^2,z^2],[x^4,y^4,z^4]|=|[x^2,y^2,z^2],[x^4,y^4,z^4],[x, y, z]| =x y z (x-y)(y-z)(z-x)(x+y+z)