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

Show that (x + omega y + omega^(2)z) (x + omega^(2)y + omega z) = x^(2) + y^(2) + z^(2)- yz - zx - xy

If 1 , omega , omega^(2) are the cube roots of unity, then prove that ( x + y + z) ( x + y omega + z omega^(2)) ( x + yomega^(2) + z omega) = x^(3) + y^(3) + z^(3) - 3xyz

If omega be an imaginary cube root or unity, prove that (x+y omega+ z omega^(2))^(4)+ (x omega+ y omega^(2)+z)^(4)+(x omega^(2)+y+ z omega)^(4)=0

Prove that x^3+y^3+z^3-3xyz =(x+y+z)(x+omegay+omega^2z)(x+yomega^2+z omega)

If 1,omega,omega^2 are the cube roots of unity then prove the following i) (1-omega+omega^2) + (1+omega-omega^2)^5=32 ii) (x+y+z)(x+yomega+zomega^2)(x+yomega^2+zomega)=x^3+y^3+z^3-3xyz

(x omega ^ (2) + y omega + z) / (x omega + y + omega ^ (2)) = ((x omega + y + z omega ^ (2)) / (x omega ^ (2) + and omega + z)) ^ (2)

If x=a+b, y=a omega+b omega^(2), z=a omega^(2)+b omega , then x y z=

If x=a+b,y=a omega+b omega^(2),z=a omega^(2)+b omega prove that x^(3)+y^(3)+z^(3)=3(a^(3)+b^(3))

If x= a + b, y= a omega^(2) + b omega, z= a omega + b omega^(2) , then show that x^(3)+ y^(3) + z^(3)= 3(a^(3) + b^(3))