Prove Theorem 1: Let $n \in \mathbb { N }$ be a natural integer and let $P \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$ be monic. Let $\lambda \in \mathbb { R }$ be a root of $P _ { \mid t = 0 }$ of multiplicity $d$. Then there exists $r \in \mathbb { R } _ { + } ^ { * }$ such that $r \leqslant \rho$ and $F \in \mathscr { D } _ { r } \left( \mathbb { R } _ { d } [ X ] \right)$ and $G \in \mathscr { D } _ { r } \left( \mathbb { R } _ { n - d } [ X ] \right)$ monic such that $P = F G$ and $F _ { \mid t = 0 } = ( X - \lambda ) ^ { d }$.
Prove Theorem 1: Let $n \in \mathbb { N }$ be a natural integer and let $P \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$ be monic. Let $\lambda \in \mathbb { R }$ be a root of $P _ { \mid t = 0 }$ of multiplicity $d$. Then there exists $r \in \mathbb { R } _ { + } ^ { * }$ such that $r \leqslant \rho$ and $F \in \mathscr { D } _ { r } \left( \mathbb { R } _ { d } [ X ] \right)$ and $G \in \mathscr { D } _ { r } \left( \mathbb { R } _ { n - d } [ X ] \right)$ monic such that $P = F G$ and $F _ { \mid t = 0 } = ( X - \lambda ) ^ { d }$.