When does the polynomial $1 + x + \cdots + x ^ { n }$ have $x - a$ as a factor? Here $n$ is a positive integer greater than 1000 and $a$ is a real number.
(A) if and only if $a = - 1$
(B) if and only if $a = - 1$ and $n$ is odd
(C) if and only if $a = - 1$ and $n$ is even
(D) We cannot decide unless $n$ is known.

Let $f ( x ) = 7 x ^ { 32 } + 5 x ^ { 22 } + 3 x ^ { 12 } + x ^ { 2 }$. (i) Find the remainder when $f ( x )$ is divided by $x ^ { 2 } + 1$. (ii) Find the remainder when $x f ( x )$ is divided by $x ^ { 2 } + 1$. In each case your answer should be a polynomial of the form $a x + b$, where $a$ and $b$ are constants.

Find a polynomial $p(x)$ that simultaneously has both the following properties.
(i) When $p(x)$ is divided by $x^{100}$ the remainder is the constant polynomial 1.
(ii) When $p(x)$ is divided by $(x-2)^{3}$ the remainder is the constant polynomial 2.

A function $g$ satisfies the property that $g(k) = 3k + 5$ for each of the 15 integer values of $k$ in $[1,15]$.
For each statement below, state if it is true or false.
(i) If $g(x)$ is a linear polynomial, then $g(x) = 3x + 5$.
(ii) $g$ cannot be a polynomial of degree 10.
(iii) $g$ cannot be a polynomial of degree 20.
(iv) If $g$ is differentiable, then $g$ must be a polynomial.

Consider the polynomial $$p(x) = x^6 + 10x^5 + 11x^4 + 12x^3 + 13x^2 - 12x - 11.$$ Find the remainder when $p(x)$ is divided by $x+1$. [1 point]

For a cubic function $f ( x )$ with leading coefficient 1 and a function $g ( x )$ that is continuous on the set of all real numbers, the following conditions are satisfied. (가) For all real numbers $x$, $f ( x ) g ( x ) = x ( x + 3 )$. (나) $g ( 0 ) = 1$ When $f ( 1 )$ is a natural number, what is the minimum value of $g ( 2 )$? [4 points]
(1) $\frac { 5 } { 13 }$
(2) $\frac { 5 } { 14 }$
(3) $\frac { 1 } { 3 }$
(4) $\frac { 5 } { 16 }$
(5) $\frac { 5 } { 17 }$

For a natural number $n$, let $a _ { n }$ be the remainder when the polynomial $2 x ^ { 2 } - 3 x + 1$ is divided by $x - n$. Find the value of $\sum _ { n = 1 } ^ { 7 } \left( a _ { n } - n ^ { 2 } + n \right)$. [3 points]

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

For a nonzero polynomial $P \in \mathbb{R}_n[X]$, express $\operatorname{deg}(\tau(P))$ and $\operatorname{cd}(\tau(P))$ in terms of $\operatorname{deg}(P)$ and $\operatorname{cd}(P)$.

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

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 a non-constant polynomial $P \in \mathbb{R}_n[X]$, express $\operatorname{deg}(\delta(P))$ and $\operatorname{cd}(\delta(P))$ in terms of $\operatorname{deg}(P)$ and $\operatorname{cd}(P)$.

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 $P_\lambda$.

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 $\mathbb{C}$, $Q_\lambda(1) = \lambda P'(\lambda)$.

Let $n$ be a non-zero natural number. We consider the polynomial $R(X) = X^{2n} + 1$. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$.
Show that $$R(X) = \prod_{k=1}^{2n}(X - \omega_k)$$

Let $n$ be a non-zero natural number. We consider the polynomial $R(X) = X^{2n} + 1$. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$. Show that $$R(X) = \prod_{k=1}^{2n}(X - \omega_k)$$

Show that there exist $K$ polynomials $L_{1}, \ldots, L_{K}$ in $\mathbb{R}_{K-1}[X]$ such that, for any function $f \in \mathcal{C}^{K}([0,1])$, the polynomial $P = \sum_{j=1}^{K} f\left(x_{j}\right) L_{j}$ satisfies $$\forall \ell \in \llbracket 1, K \rrbracket, \quad P\left(x_{\ell}\right) = f\left(x_{\ell}\right).$$

Show that there exist $K$ polynomials $L_1, \ldots, L_K$ in $\mathbb{R}_{K-1}[X]$ such that, for any function $f \in \mathcal{C}^K([0,1])$, the polynomial $P = \sum_{j=1}^K f\left(x_j\right) L_j$ satisfies $$\forall \ell \in \llbracket 1, K \rrbracket, \quad P\left(x_\ell\right) = f\left(x_\ell\right).$$

Let $n \in \mathbb{N}^*$ and $$T _ { n } ( X ) = \sum _ { p = 0 } ^ { \lfloor n / 2 \rfloor } ( - 1 ) ^ { p } \binom { n } { 2 p } X ^ { n - 2 p } \left( 1 - X ^ { 2 } \right) ^ { p }.$$ Show that $T _ { n }$ is a polynomial of degree $n$. Explicitly state the leading coefficient of $T _ { n }$.