= Solution
The <Chebyshev inequality> states that for a random variable $Y$ with finite variance and every $\delta>0$,
$$
\mathbb P(|Y-\mathbb EY|\geq\delta)
\leq\frac{\operatorname{var}Y}{\delta^2}.
$$
Indeed, <Markov inequality> applied to the nonnegative variable $(Y-\mathbb EY)^2$ gives
$$
\mathbb P((Y-\mathbb EY)^2\geq\delta^2)
\leq\frac{\mathbb E[(Y-\mathbb EY)^2]}{\delta^2}.
$$
The sum $S_n=X_1+\cdots+X_n$ has the <binomial distribution>, so
$$
\boxed{
B_n(p)=
\sum_{k=0}^n f(k/n)\binom nkp^k(1-p)^{n-k}}.
$$
For fixed $n$ this is a finite sum of polynomial functions of $p$, hence is a polynomial.
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