Let $n$ be a non-zero natural number. We denote $E _ { n } = \mathbb { R } _ { n } [ X ]$ and for all $k \in \llbracket 0 , n \rrbracket , P _ { k } = X ^ { k }$.

Let $\alpha$ be a real number.

\begin{enumerate}
  \item Justify that the family $\mathcal { E } = \left( 1 , X - \alpha , \ldots , ( X - \alpha ) ^ { n } \right)$ is a basis of $E _ { n }$.
  \item Let $P$ be a polynomial in $E _ { n }$. Give without proof the decomposition of $P$ in the basis $\mathcal { E }$ using the successive derivatives of the polynomial $P$.
  \item Suppose that $\alpha$ is a root of order $r \in \llbracket 1 , n \rrbracket$ of $P$. Determine the quotient and remainder of the Euclidean division of $P$ by $( X - \alpha ) ^ { r }$.
\end{enumerate}

To every polynomial $P$ of $E _ { n }$, we associate the polynomial $Q$ defined by:
$$Q ( X ) = X P ( X ) - \frac { 1 } { n } \left( X ^ { 2 } - 1 \right) P ^ { \prime } ( X )$$
and we denote by $T$ the application that associates $Q$ to $P$.

\begin{enumerate}
  \setcounter{enumi}{3}
  \item Let $k \in \llbracket 0 , n \rrbracket$. Determine $T \left( P _ { k } \right)$.
  \item Show that $T$ is an endomorphism of $E _ { n }$.
  \item Write the matrix $M$ of $T$ in the basis $\mathscr { B } = \left( P _ { 0 } , P _ { 1 } , \ldots , P _ { n } \right)$ of $E _ { n }$.
  \item Suppose that $\lambda$ is a real eigenvalue of the endomorphism $T$ and let $P$ be a monic polynomial, eigenvector associated with the eigenvalue $\lambda$.
  \begin{enumerate}
    \item[7.1.] Show that $P$ has degree $n$.
    \item[7.2.] Let $z _ { 0 }$ be a complex root of $P$ with multiplicity order $r \in \mathbb { N } ^ { * }$. Prove that $z _ { 0 } ^ { 2 } - 1 = 0$.
    \item[7.3.] Deduce an expression for $P$.
  \end{enumerate}
  \item Determine the eigenvectors of the endomorphism $T$. Is the endomorphism $T$ diagonalisable?
\end{enumerate}