Let $P \in \mathbb{R}_n[X]$. For $k \in \mathbb{N}$, give the expression of $\tau^k(P)$ as a function of $P$.

Let $n$ be a non-zero natural number. Let $P$ be in $\mathbb{C}_{2n}[X]$, and, for all $\lambda \in \mathbb{C}$, $P_\lambda(X) = P(\lambda X) - P(\lambda)$. For all $\lambda$ in $\mathbb{C}$, we denote by $Q_\lambda$ the quotient of $P_\lambda$ by $X - 1$: $$Q_\lambda(X) = \frac{P(\lambda X) - P(\lambda)}{X - 1} \in \mathbb{C}_{2n-1}[X]$$
Show that, for all $\lambda$ in $\mathbb{C}$, $Q_\lambda(1) = \lambda P'(\lambda)$.

Let $n$ be a non-zero natural number. Let $P$ be in $\mathbb{C}_{2n}[X]$, and for all $\lambda$ in $\mathbb{C}$, let $Q_\lambda$ be the quotient of $P_\lambda = P(\lambda X) - P(\lambda)$ by $X-1$: $$Q_\lambda(X) = \frac{P(\lambda X) - P(\lambda)}{X - 1} \in \mathbb{C}_{2n-1}[X]$$ Show that, for all $\lambda$ in $\mathbb{C}$, $Q_\lambda(1) = \lambda P'(\lambda)$.

The third-degree polynomial $\mathrm { P } ( \mathrm { x } )$ with leading coefficient 1 is divisible without remainder by $x ^ { 2 } + 4$. The remainder obtained from dividing the polynomial $P ( 2 x )$ by $2 x - 3$ is 52.
Accordingly, what is the value of $\mathbf { P } ( 2 )$?
A) 20
B) 22
C) 24
D) 26
E) 28

$$P ( x ) = x ^ { 3 } - m x + 1$$
The remainder when $P ( x - 1 )$ is divided by $x + 1$ equals the remainder when $P ( x + 1 )$ is divided by $x - 1$.
Accordingly, what is m?
A) 2
B) 4
C) 6
D) - 1
E) - 8

Let $P(x)$ be a polynomial. A number $a$ satisfying the equation $P(a) = 0$ is called a root of this polynomial. For polynomials $P(x)$ and $R(x)$
$$\begin{aligned} &\mathrm{P}(\mathrm{x}) = \mathrm{x}^{2} - 1 \\ &\mathrm{R}(\mathrm{x}) = \mathrm{P}(\mathrm{P}(\mathrm{x})) \end{aligned}$$
the following equations are given.
Accordingly,
I. $-1$ II. $0$ III. $1$
which of these numbers are roots of the polynomial $\mathbf{R}(\mathbf{x})$?
A) Only I
B) Only II
C) Only III
D) I and III
E) II and III