<s>[INST] Cyclic polynomial proof If$$  x+y+z = 0 $$
Then prove, 
$$ (x^2+xy+y^2)^3+(y^2+yz+z^2)^3+(z^2+zx+x^2)^3$$
$$=3(x^2+xy+y^2)(y^2+yz+z^2)(z^2+zx+x^2)$$
 [/INST] HINT:
Observe that if we put  $x^2+xy+y^2=a$ etc., we need to prove $a^3+b^3+c^3=3abc$
From this, the above proposition will be true 
either if $a+b+c=0$ 
or if $a=b=c$ for real $a,b,c$
Now, $x^2+xy+y^2-(y^2+yz+z^2)=x^2-z^2+xy-yz=(x-z)(x+z+y)=0 $
Alternatively eliminating $x,$
$x^2+xy+y^2=x(x+y)+z^2=\{-(y+z)\}(-z)+y^2$ as $x+y+z=0$
$\implies x^2+xy+y^2=y^2+yz+z^2$
Similarly we can prove, $ x^2+xy+y^2=z^2+zx+x^2 $
$$\implies x^2+xy+y^2=y^2+yz+z^2=z^2+zx+x^2$$
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