# Polynomial ring/Field/Replacement/Structure/Exercise

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}K[X]}$ be the polynomial ring over ${\displaystyle {}K}$. Let ${\displaystyle {}a\in K}$. Prove that the evaluating function
${\displaystyle \psi \colon K[X]\longrightarrow K,P\longmapsto P(a),}$
satisfies the following properties (here let ${\displaystyle {}P,Q\in K[X]}$).
1. ${\displaystyle {}(P+Q)(a)=P(a)+Q(a)\,.}$
2. ${\displaystyle {}(P\cdot Q)(a)=P(a)\cdot Q(a)\,.}$
3. ${\displaystyle {}1(a)=1\,.}$