A monic polynomial of degree $d \geq 1$ $$P = \sum _ { i = 0 } ^ { d } a _ { i } X ^ { i } \in \mathbb { C } [ X ]$$ is called reciprocal if $a _ { i } = a _ { d - i }$ for $0 \leq i \leq d$. (a) Show that a monic polynomial $P \in \mathbb { C } [ X ]$ of degree $d$ is reciprocal if and only if $X ^ { d } P \left( \frac { 1 } { X } \right) = P$. (b) Let $P \in \mathbb { C } [ X ]$ be a monic reciprocal polynomial. Show that if $x \in \mathbb { C }$ is a root of $P$, then $x \neq 0$ and $\frac { 1 } { x }$ is also a root of $P$, with the same multiplicity.
A monic polynomial of degree $d \geq 1$
$$P = \sum _ { i = 0 } ^ { d } a _ { i } X ^ { i } \in \mathbb { C } [ X ]$$
is called reciprocal if $a _ { i } = a _ { d - i }$ for $0 \leq i \leq d$.
(a) Show that a monic polynomial $P \in \mathbb { C } [ X ]$ of degree $d$ is reciprocal if and only if $X ^ { d } P \left( \frac { 1 } { X } \right) = P$.\\
(b) Let $P \in \mathbb { C } [ X ]$ be a monic reciprocal polynomial. Show that if $x \in \mathbb { C }$ is a root of $P$, then $x \neq 0$ and $\frac { 1 } { x }$ is also a root of $P$, with the same multiplicity.