Is it possible to factor $x^3+y^3+z^3$

Question

It's a quick question: is it possible to factor $$x^3+y^3+z^3$$ into a form like $(x+y+z)(\ldots)$?

Wolfram Alpha says... no. – Parcly Taxel Oct 23 '16 at 08:29 — Parcly Taxel, Oct 23 '16 at 08:29

score 6 · Answer 1 · edited Apr 13 '17 at 12:20

6

The polynomial $x^3+y^3+z^3$ is irreducible in $\mathbf{C}[x,y,z]$. More generally, so is $x^n+y^n+z^n$ for $n \geq 1$. See the answers given to this question.

edited Apr 13 '17 at 12:20

Community

1

answered Oct 23 '16 at 08:45

BrL

1,286

score 5 · Answer 2 · answered Oct 23 '16 at 08:40

5

As an alternative to the other answer, note that $(x + y + z)(...)$ has a zero at $2,2,-4$, while $2^3 + 2^3 + (-4)^3 = -48$.

answered Oct 23 '16 at 08:40

Mees de Vries

26,947

score 2 · Answer 3 · answered Oct 23 '16 at 08:47

General factorization over polynomials is not possible. But factorization is possible in special cases under Diophantine solution i.e. if $x,y$ and $z$ are positive integers such that $3xyz$ is divisible by $x+y+z$ then factorization is possible.

It is possible to creeate infinitely many such Diophantine solution in parametric form. For example $x = p, y = 2p, z = 3p$.

Is it possible to factor $x^3+y^3+z^3$

3 Answers3