Let $R$ be a ring. The polynomial ring over $R$ is a subring of $R[[X]]$:
\[R[X] = \{p \in R[[X]] : (\exists d \in \N \, \forall n > d : p_n = 0)\}.\]Each element of $R[X]$ is called a polynomial over $R$.
$0 \in R[X]$ is the zero polynomial.
The degree of a polynomial $p \in R[X]$ is
\[\deg p = \begin{cases} -\infty, & \text{if $p = 0$}; \\ \max\{d \in \N : p_d \neq 0\}, & \text{otherwise}. \end{cases}\]The constant term of a polynomial $p\in R[X]$ is
\[p_0.\]The leading coefficient of a non-zero polynomial $p \in R[X]$ is
\[p_{\deg p}.\]A non-zero polynomial is monic if its leading coefficient is $1$.
Let $R$ be a ring. For any $p$, $q \in R[X]$,
\[\deg(p + q) \leq \max(\deg p, \deg q).\]
Let $R$ be an integral domain. For any $p$, $q \in R[X]$,
\[\deg(pq) = \deg p + \deg q.\]