Prove that for every polynomial $P(x)$ with real coefficients there exist a positive integer $m$ and polynomials $P_{1}(x),\ldots , P_{m}(x)$ with real coefficients such that \[P(x) = (P_{1}(x))^{3}+\ldots +(P_{m}(x))^{3}\]
Source: Baltic Way 2006
Tags: algebra, polynomial, algebra unsolved
Prove that for every polynomial $P(x)$ with real coefficients there exist a positive integer $m$ and polynomials $P_{1}(x),\ldots , P_{m}(x)$ with real coefficients such that \[P(x) = (P_{1}(x))^{3}+\ldots +(P_{m}(x))^{3}\]