(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that if x^{3}+x^{2}y+xy^{2}+y^{3}= 0, then x = y = 0 or x = -y.

2. Relevant equations

N/A

3. The attempt at a solution

Assume that x^{3}+x^{2}y+xy^{2}+y^{3}= 0, in which case, it follows that x^{3}+y^{3}= -(x^{2}y+xy^{2}) or (x+y)(x^{2}-xy+y^{2}) = -xy(x+y). Equality clearly holds if x+y = 0. Now, suppose that x+y =/= 0, and divide through by x+y. This leaves the equality x^{2}-xy+y^{2}= -xy or x^{2}+y^{2}= 0, which can only happen if x = y = 0. Therefore, if x^{3}+x^{2}y+xy^{2}+y^{3}= 0, then x = -y or x = y = 0.

Does this 'proof' work?

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# Zeros proof math

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