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. $\mathrm{deg}(P+Q)\le max\{\mathrm{deg}(P),\mathrm{deg}(Q)\},$

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

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