Show that, for all $x \in \mathbb { R }$ and for all $n \in \mathbb { N } ^ { * }$, $$\left( x ^ { n } - 1 \right) ^ { 2 } = \prod _ { k = 1 } ^ { n } \left( x ^ { 2 } - 2 x \cos \frac { 2 k \pi } { n } + 1 \right)$$

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ Deduce the kernel $\operatorname{ker}(\delta)$ and the image $\operatorname{Im}(\delta)$ of the endomorphism $\delta$.

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ More generally, for $j \in \llbracket 1, n \rrbracket$, show the following equalities: $$\operatorname{ker}\left(\delta^j\right) = \mathbb{R}_{j-1}[X] \quad \text{and} \quad \operatorname{Im}\left(\delta^j\right) = \mathbb{R}_{n-j}[X]$$

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ For $k \in \mathbb{N}$ and $P \in \mathbb{R}_n[X]$, express $\delta^k(P)$ in terms of $\tau^j(P)$ for $j \in \llbracket 0, k \rrbracket$.

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ In this question, we seek all vector subspaces of $\mathbb{R}_n[X]$ stable under the application $\delta$.
a) For a nonzero polynomial $P$ of degree $d \leqslant n$, show that the family $(P, \delta(P), \ldots, \delta^d(P))$ is free. What is the vector space spanned by this family?
b) Deduce that if $V$ is a vector subspace of $\mathbb{R}_n[X]$ stable under $\delta$ and not reduced to $\{0\}$, there exists an integer $d \in \llbracket 0, n \rrbracket$ such that $V = \mathbb{R}_d[X]$.

Let $P , Q \in \mathbb { R } [ X ]$ be nonzero polynomials of respective degrees $p$ and $q$ strictly positive. Show that the linear map $L _ { P , Q }$ defined by $$\left\lvert \, \begin{array} { c c c } L _ { P , Q } : \quad \mathbb { R } _ { q - 1 } [ X ] \times \mathbb { R } _ { p - 1 } [ X ] & \rightarrow \quad \mathbb { R } _ { p + q - 1 } [ X ] \\ ( V , W ) & \mapsto V P + W Q \end{array} \right.$$ is an isomorphism if and only if $P$ and $Q$ are coprime in $\mathbb { R } [ X ]$.

Let $n$ be a non-zero natural number. Let $P$ be in $\mathbb{C}_{2n}[X]$, and, for all $\lambda \in \mathbb{C}$, $P_\lambda(X) = P(\lambda X) - P(\lambda)$.
If $\lambda \in \mathbb{C}$, verify that $X - 1$ divides $P_\lambda$.

Let $n$ be a non-zero natural number. Let $P$ be in $\mathbb{C}_{2n}[X]$, and, for all $\lambda \in \mathbb{C}$, $P_\lambda(X) = P(\lambda X) - P(\lambda)$. If $\lambda \in \mathbb{C}$, verify that $X - 1$ divides $P_\lambda$.

We denote by $\Delta_{d}$ the endomorphism of $\mathbb{K}_{d}[X]$ induced by $\Delta$, where $\Delta(P) = P(X+1) - P(X)$. Let $d \in \mathbb{N}^{*}$. Determine an annihilating polynomial of $\Delta_{d}$. Is the endomorphism $\Delta_{d}$ diagonalisable?