Alternatively, if you simply want a reference, this is an application of the highly nontrivial Schmüdgen theorem (a.k.a. Schmüdgen's Positivstellensatz) [1]. The theorem says that if you have a compact semialgebraic set $K \subset \mathbb{R}^n$ defined by inequalities $\{g_1 \geq 0, \dots, g_s \geq 0\}$, then any *positive* polynomial $f$ on $K$ belongs to the cone generated by $\{g_1, \dots, g_s\}$, i.e. sums with positive coefficients of products of the $g_i$'s (including powers) together with squares of arbitrary polynomials.

In your case, you have $g_1(x)=x$ and $g_2(x)=1-x$, and because you're in the one variable case, you should be able to get rid of the arbitrary squares (I might come back to this answer when I have more time).


[1] K. Schmüdgen, The K-moment problem for compact semi-algebraic sets. Math. Ann. 289 (1991), no. 2, 203-206.