MathNet 2009-part-04 — 175 problems (EN–VI) Question 102

Find all such positive integer $n$ that for every polynomial $g(x)$ with real coefficients there exist such polynomials $p_1(x), p_2(x),..., p_n(x)$ and $q_1(x), q_2(x),..., q_n(x)$ that $g(x) = \sum_{i=1}^{n} (p_i^2(x) + q_i^3(x))$.