Polynomial ring/Field/Degree/Simple rules/Fact/Proof/Exercise

Let ${}K$ be a field and let ${}K[X]$ be the polynomial ring over ${}K$. Prove the following properties concerning the degree of a polynomial:

1. ${}\operatorname {deg} \,(P+Q)\leq \max\{\operatorname {deg} \,(P),\,\operatorname {deg} \,(Q)\}\,,$

2. ${}\operatorname {deg} \,(P\cdot Q)=\operatorname {deg} \,(P)+\operatorname {deg} \,(Q)\,.$

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