Algebra problem

Question

If x+y+z=0 then prove that (x²+xy+y²)³+(y²+yz+z²)³+(z²+zx+x²)³=3(x²+xy+y²)(y²+yz+z²)(z²+zx+x²)

Answer 1

Given: x+y+z=0

To prove: (x²+xy+y²)³+(y²+yz+z²)³+(z²+zx+x²)³=3(x²+xy+y²)(y²+yz+z²)(z²+zx+x²)

Let a = x²+xy+y²

b = y²+yz+z² and

c = z²+zx+x²

Now putting the value in above equation we get

a³+b³+c³ = 3abc

We know the identity

Hence, (x²+xy+y²)³+(y²+yz+z²)³+(z²+zx+x²)³=3(x²+xy+y²)(y²+yz+z²)(z²+zx+x²)

Proved.