Let $P$ be a polynomial of degree $p$ written in factored form $P = a_p \prod_{i=1}^{d} (X - \lambda_i)^{m_i}$, where $\lambda_1, \ldots, \lambda_d$ are the distinct complex roots of $P$ and $m_1, \ldots, m_d$ their multiplicities.
Write in factored form the polynomial $X^p P\left(\frac{1}{X}\right)$ and prove that if $P$ is reciprocal then for every integer $i$, $1 \leq i \leq d$, $\lambda_i$ is nonzero and $\frac{1}{\lambda_i}$ is a root of $P$ with multiplicity $m_i$.