Let $P$ be a real polynomial, $a$ a real number. What is the remainder of the Euclidean division of $P$ by $(X-a)$?
The remainder of the Euclidean division of $P$ by $(X-a)$ is of degree at most 0, so it is a constant denoted by $C$. There thus exists a polynomial $Q$ with real coefficients such that $P(X)=Q(X)(X-$ $a)+C$. By evaluating at $a$, we obtain $C=P(a)$. Therefore, the remainder of the Euclidean division is $P(a)$.