Let $R$ be a commutative ring and let $f, g \in R[x]$ be nonzero polynomials. Then:
(i) $\deg(f + g) \le \max(\deg f, \deg g)$.
(ii) $\deg(fg) \le \deg f + \deg g$.
(iii) If $R$ is an integral domain, then $\deg(fg) = \deg f + \deg g$.
In particular, if $R$ is an integral domain, then $R[x]$ is also an integral domain.