In this paper, we are interested in the following problem: determine all \(f \in {\mathbb Z}[x]\) which are expressible as a sum of several cubes of polynomials in \({\mathbb Z}[x]\) . We establish a necessary and sufficient condition in terms of the coefficients of f for this to occur. Similar conditions for f to be expressible as a sum of pth powers of polynomials in \({\mathbb Z}[x]\) are established not only for \(p=3\) , but also for \(p=5\) , \(p=7\) and \(p=11\) . In the latter case, the necessary and sufficient condition is a simple one: the polynomial \(f(x)=a_0+a_1x+\dots +a_nx^n \in {\mathbb Z}[x]\) is expressible as a sum of the eleventh powers of polynomials in \({\mathbb Z}[x]\) if and only if \(11 \mid ka_k\) for each \(k \in \{0,1,\dots ,n\}\) .
