Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If$$ x+y+z = 0 $$ Then prove, $$ (x^2+xy+y^2)^3+(y^2+yz+z^2)^3+(z^2+zx+x^2)^3$$ $$=3(x^2+xy+y^2)(y^2+yz+z^2)(z^2+zx+x^2)$$

share|cite|improve this question

1 Answer 1

up vote 4 down vote accepted


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 $

$$\implies x^2+xy+y^2=y^2+yz+z^2=z^2+zx+x^2$$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.