Let $A , B \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$. We assume that $B$ is monic of degree $d \leqslant n$. 9a. Show that there exist elements $Q \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n - d } [ X ] \right)$ and $R \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { d - 1 } [ X ] \right)$ uniquely determined such that $A = B Q + R$. The elements $Q$ and $R$ are called respectively the quotient and the remainder of the Euclidean division of $A$ by $B$. 9b. Let furthermore $r , s \in \mathbb { R } _ { + } ^ { * }$ with $r < \rho$. Show that, if $\left\| B - X ^ { d } \right\| _ { r , s } < s ^ { d }$, then $$\| Q \| _ { r , s } \leqslant \frac { \| A \| _ { r , s } } { s ^ { d } - \left\| B - X ^ { d } \right\| _ { r , s } } \quad \text { and } \quad \| R \| _ { r , s } \leqslant \frac { s ^ { d } \cdot \| A \| _ { r , s } } { s ^ { d } - \left\| B - X ^ { d } \right\| _ { r , s } }$$ (One may start by treating the case where $B = X ^ { d }$.)
Let $A , B \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$. We assume that $B$ is monic of degree $d \leqslant n$.
\textbf{9a.} Show that there exist elements $Q \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n - d } [ X ] \right)$ and $R \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { d - 1 } [ X ] \right)$ uniquely determined such that $A = B Q + R$.
The elements $Q$ and $R$ are called respectively the quotient and the remainder of the Euclidean division of $A$ by $B$.
\textbf{9b.} Let furthermore $r , s \in \mathbb { R } _ { + } ^ { * }$ with $r < \rho$. Show that, if $\left\| B - X ^ { d } \right\| _ { r , s } < s ^ { d }$, then
$$\| Q \| _ { r , s } \leqslant \frac { \| A \| _ { r , s } } { s ^ { d } - \left\| B - X ^ { d } \right\| _ { r , s } } \quad \text { and } \quad \| R \| _ { r , s } \leqslant \frac { s ^ { d } \cdot \| A \| _ { r , s } } { s ^ { d } - \left\| B - X ^ { d } \right\| _ { r , s } }$$
(One may start by treating the case where $B = X ^ { d }$.)