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

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}K[X]}$ be the polynomial ring over ${\displaystyle {}K}$. Prove the following properties concerning the degree of a polynomial:
1. ${\displaystyle {}\operatorname {deg} \,(P+Q)\leq \max\{\operatorname {deg} \,(P),\,\operatorname {deg} \,(Q)\}\,,}$
2. ${\displaystyle {}\operatorname {deg} \,(P\cdot Q)=\operatorname {deg} \,(P)+\operatorname {deg} \,(Q)\,.}$