Chapter 8.1, Problem 10E

Let $R$ be a commutative ring with unity. Prove that

$\mathrm{deg}\left(f\left(x\right)g\left(x\right)\right)\le \mathrm{deg}f\left(x\right)+\mathrm{deg}g\left(x\right)$

for all nonzero $f\left(x\right)$, $g\left(x\right)$ in $R\left[x\right]$, even if $R$ in not an integral domain.