सिद्ध कीजिए कि `|{:(x+y,y,z),(y+z,z,x),(z+x,x,y):}|=3xyz-x^(3)-y^(3)-z^(3)`

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)]

Using properties of determinants, prove that |{:(y + z ,z + x ,x + y ),(z + x ,x + y ,y + z),(x + y ,y + z,z + x ):}|=2 |{:(x, y, z),(y, z, x),(z, x, y):}|= - 2 (x^(3) + y^(3) + z^(3) - 3xyz)

Prove that |{:(y+z,x,x),(y,z+x,y),(z,z,x+y):}|=4xyz

prove that: |(y+z,z,y),(z,z+x,x),(y,x,x+y)|=4xyz

prove that: |(y+z,z,y),(z,z+x,x),(y,x,x+y)|=4xyz

|{:(y+z," "z," "y),(" "z,z+x," "x),(" "y," "x,x+y):}|=4xyz

Factorise : (x+y+z)(x^(3)+y^(3)+z^(3)+xyz)-x^(4)-y^(4)-z^(4)

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