Polynomial ring/Evaluation/Remark

Let ${\displaystyle {}K}$ be a field, let ${\displaystyle {}K[X]}$ be the polynomial ring over ${\displaystyle {}K}$ and ${\displaystyle {}a\in K}$. Then the evaluation mapping

${\displaystyle K[X]\longrightarrow K,P\longmapsto P(a),}$

is ${\displaystyle {}K}$-linear. Moreover, we have

${\displaystyle {}(PQ)(a)=P(a)Q(a)\,,}$

see exercise.