ELMO SL 2018 A4

Elmo calls a monic polynomial with real coefficients tasty if all of its coefficients are in the range $[-1,1]$ . A monic polynomial $P$ with real coefficients and complex roots $\chi_1,\cdots,\chi_m$ (counted with multiplicity) is given to Elmo, and he discovers that there does not exist a monic polynomial $Q$ with real coefficients such that $PQ$ is tasty. Find all possible values of $\max\left(|\chi_1|,\cdots,|\chi_m|\right)$ .

Proposed by Carl Schildkraut