Let $\widehat { \mu } = \left( \mu _ { 1 } > \cdots > \mu _ { n } \right) \in \mathbb { R } ^ { n }$. We are given integers $m _ { k } \geqslant 1$ for $k = 1 , \ldots , n$. We set $$Q _ { 1 } = \prod _ { k = 1 } ^ { n } \left( X - \mu _ { k } \right) ^ { m _ { k } } \quad \text { and, this time, } \quad P _ { j } = \frac { Q _ { 1 } } { X - \mu _ { j } } .$$ Show that $$Q _ { 1 } \wedge Q _ { 1 } ^ { \prime } = \prod _ { k = 1 } ^ { n } \left( X - \mu _ { k } \right) ^ { m _ { k } - 1 }$$

We still assume that $p$ is an integer greater than or equal to 2. Let $R \in \mathbb { C } [ X ]$ be a polynomial of degree greater than or equal to 1 with simple roots. Prove that the polynomial $R \left( X ^ { p } \right)$ has simple roots if and only if $R ( 0 ) \neq 0$.

We denote by $n$ the integer part of $\frac{N}{2}$. We continue the study of the polynomial $R_{N}$ (the even polynomial in $B_N$ minimising $L$, with all roots in $[-1,1]$).
Show that $R_{N}$ is the square of a polynomial: $R_{N}(X) = U_{N}(X)^{2}$ where $U_{N}(1) = 1$ and $U_{N}(-1) = \pm 1$. What can we say about the parity of $U_{N}$?

We consider a natural integer $n$ and a complex number $a$. We define a family of polynomials $(A_0, A_1, \ldots, A_n)$ by setting $$A_0 = 1 \quad \text{and, for all } k \in \llbracket 1, n \rrbracket, \quad A_k = \frac{1}{k!} X(X - ka)^{k-1}.$$ Prove that for all $k \in \llbracket 1, n \rrbracket$, $A_k'(X) = A_{k-1}(X - a)$.

Until the end of part A, we assume that all roots of $p$ are stable and have multiplicity 1.
Verify that $p'$ is split over $\mathbf{R}$ then show that $h \wedge h_0 = 1$ and deduce that $p'$ has no stable root.

The number of different values of $a$ for which the equation $x ^ { 3 } - x + a = 0$ has two identical real roots is
(A) 0 .
(B) 1 .
(C) 2 .
(D) 3 .

2. For ALL APPLICANTS.

Let $a$ and $b$ be real numbers. Consider the cubic equation
$$x ^ { 3 } + 2 b x ^ { 2 } - a ^ { 2 } x - b ^ { 2 } = 0$$
(i) Show that if $x = 1$ is a solution of ( $*$ ) then
$$1 - \sqrt { 2 } \leqslant b \leqslant 1 + \sqrt { 2 }$$
(ii) Show that there is no value of $b$ for which $x = 1$ is a repeated root of ( $*$ ).
(iii) Given that $x = 1$ is a solution, find the value of $b$ for which $( * )$ has a repeated root. For this value of $b$, does the cubic
$$y = x ^ { 3 } + 2 b x ^ { 2 } - a ^ { 2 } x - b ^ { 2 }$$
have a maximum or minimum at its repeated root?