Problem #PRU-116704

Problem

The teacher wrote on the board in alphabetical order all possible $2^n$ words consisting of $n$ letters A or B. Then he replaced each word with a product of $n$ factors, correcting each letter A by $x$, and each letter B by $(1-x)$, and added several of the first of these polynomials in $x$. Prove that the resulting polynomial is either a constant or increasing function in $x$ on the interval $[0,1]$.