(Case of simple roots)

Let $n \geqslant 2$. Determine the remainder of the Euclidean division of $X^{n}$ by $X^{2}-3 X+2$.
We write the Euclidean division of $X^{n}$ by $X^{2}-3 X+2$. There then exists a (unique) pair $(Q, R)$ such that:

$$
X^{n}=\left(X^{2}-3 X+2\right) Q+R \quad \text { with } \operatorname{deg} R<\operatorname{deg} B=2 \text { i.e. } R=a X+b \text { with }(a, b) \in \mathbb{K}^{2}
$$

Evaluating this relation at the roots of $X^{2}-3 X+2$, namely 1 and 2, allows us to establish a system of two equations in $a$ and $b$. Solving this system then allows us to conclude that $R=\left(2^{n}-1\right) X-\left(2^{n}-2\right)$.