Multiple roots of polynomials with coefficients $\pm 1$

Question P.Can a polynomial $P(x)=\sum_{n=0}^ma_nx^n$ with coefficients $a_n\in\{-1,1\}$ (and $P(1)=0$) have a multiple root in the interval $(\tfrac12,1)$?

Also I am interested in a similar question for analytic functions.

Question A.Let $f(x)=\sum_{n=0}^\infty a_nx^n$ a series with coefficients $a_n\in\{-1,1\}$ such that $\sup_{m\in\mathbb N}|\sum_{n=0}^ma_i|<\infty$. Can the analytic function $f$ have a multiple root in the interval $(0,1)$?