Polynomial ring/Field/Replacement/Structure/Exercise

Let ${}K$ be a field and let ${}K[X]$ be the polynomial ring over ${}K$. Let ${}a\in K$. Prove that the evaluating function

$\psi \colon K[X]\longrightarrow K,P\longmapsto P(a),$

satisfies the following properties (here let ${}P,Q\in K[X]$).

1. ${}(P+Q)(a)=P(a)+Q(a)\,.$

2. ${}(P\cdot Q)(a)=P(a)\cdot Q(a)\,.$

3. ${}1(a)=1\,.$

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