Bernstein polynomial degree preservation ($\mathrm{deg}{B}_{n}p=\mathrm{deg}p(\mathrm{deg}p\le n)$)

For $K$ with a binomial distribution of parameters $n,x$, the Bernstein image of a monomial is $B_n(x^m)=\mathbb{E}(K/n{)}^m$. Expand $K^m$ in the monic falling factorial basis. Since $\mathbb{E}{K}^{\underline{r}}=n^{\underline{r}}{x}^r$, the leading coefficient becomes $n^{\underline{m}}/n^m$, which is nonzero for $m\le n$. Thus an actual degree-$m$ polynomial retains its degree, even though its lower coefficients generally change.