If `x+y+z=0` show that `x^(3)+y^(3)+z^(3)=3xyz`.

(a) If x + y + z=0, show that x ^(3) + y ^(3) + z ^(3)= 3 xyz. (b) Show that (a-b) ^(3) + (b-c) ^(3) + (c-a)^(3) =3 (a-b) (b-c) (c-a)

If u = log (x^(3) + y^(3) + z^(3) - 3xyz) , then (x + y + z)(u_(x) + u_(y) + u_(z)) is equal to

Verify that x ^(3) + y ^(3) + z ^(3) - 3xyz =1/2 (x + y + z) [(x-y)^(2) + (y-z) ^(2) + (z-x) ^(2) ]

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

Do the following Division 8x^(3)y^(3)z^(3) div 16 xyz

Verify that x^3 + y^3 + z^3- 3xyz=1/2 (x+y+z)[(x-y)^2+(y-z)^2+(z-x)^2]

Factorise 27 x ^(3) + y ^(3) + z ^(3)-9 xyz using identitiy.

If x+y+z=xyz , " then " sum(3x-x^(3))/(1-3x^(2)) =

Let x be the arithmetic mean y,z be the two geometric means between any two positive numbers. Then value of (y^(3) + z^(3))/(xyz) is