  Given $n$ polynomials $p_1, \dots, p_n$ of degree at most $n$ with
$\|p_i\|_\infty \le 1$ for $i \in [n]$, we show there exist signs $x_1, \dots,
x_n \in \{-1,1\}$ so that \[\Big\|\sum_{i=1}^n x_i p_i\Big\|_\infty <
30\sqrt{n}, \] where $\|p\|_\infty := \sup_{|x| \le 1} |p(x)|$. This result
extends the Rudin-Shapiro sequence, which gives an upper bound of $O(\sqrt{n})$
for the Chebyshev polynomials $T_1, \dots, T_n$, and can be seen as a
polynomial analogue of Spencer's "six standard deviations" theorem.
