The Degree Of A Product Of Polynomials

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Problem 52 (The Degree Of A Product Of Polynomials)

Suppose that $D$ is an integral domain, and suppose that $f(x),g(x)\in D[x]$.

1. Prove that $\deg(f(x)\cdot g(x)) = \deg(f(x))+\deg(g(x))$.
2. Then give an example of a commutative ring $R$ and two polynomials so that $\deg(f(x)\cdot g(x)) < \deg(f(x))+\deg(g(x))$.

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