#Chebychev's polynomial exercise problem

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ember elmBOT
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#

\textbf{Exercise 2}

We define a sequence $(R_k){k \in \mathbb{N}^*}$ of polynomials by:
[
R_1 = X, \quad R_2 = X^2 - 2,
]
and for every integer $k \geq 2$:
[
R{k+1} = X R_k - R_{k-1}.
]

\begin{enumerate}

\item Determine the polynomials $R_3$ and $R_4$.

\item Prove by induction that for every integer $k \geq 1$, the polynomial $R_k$ has degree $k$ and satisfies, for every $x \in \mathbb{C}^$:
[
R_k!\left(x + \frac{1}{x}\right) = x^k + \frac{1}{x^k}.
]

\textit{Hint: use the recurrence relation with $R_k$ and $R_{k-1}$.}

\item Let $a \in \mathbb{R}$. Determine, if they exist, the nonzero complex numbers $x$ such that
[
x + \frac{1}{x} = a.
]
Distinguish the three cases depending on the value of $a$.

\item In the rest of the exercise, let $Q$ be a polynomial of degree $2n$, with $n \in \mathbb{N}$, defined by
[
Q(X) = \sum_{k=0}^{2n} a_k X^k,
]
with $a_{2n} \neq 0$, and such that for every integer $k \in {0, \dots, n}$:
[
a_k = a_{2n-k}.
]

\begin{enumerate}

\item Can $Q$ be the polynomial
[
P_1(X) = X^3 + 2X^2 + 2X + 1 , ?
]

Can $Q$ be the polynomial
[
P_2(X) = 3X^4 + 7X^3 + 2X^2 + 7X + 3 , ?
]

Can $Q$ be the polynomial
[
P_3(X) = X^4 + 2X^3 + 2X^2 + 2X + 2 , ?
]

\item Prove that $0$ is not a root of $Q$.

\item We define the polynomial $\widetilde{Q}$ by
[
\widetilde{Q}(X) = a_n + \sum_{k=1}^{n} a_{n-k} R_k(X).
]

Let $x \in \mathbb{C}^$ and set
[
y = x + \frac{1}{x}.
]

Express $x^n \widetilde{Q}(y)$ in terms of $Q(x)$.

\item Deduce that $Q(x) = 0$ if and only if $\widetilde{Q}(y) = 0$.
What is the interest of this result for determining the roots of $Q$?

\item In this question, assume that $n = 3$ and that
[
Q(X) = X^6 + X^5 - 9X^4 + 2X^3 - 9X^2 + X + 1.
]

Verify that
[
\widetilde{Q}(X) = X^3 + X^2 - 12X.
]

\item Deduce the roots of $\widetilde{Q}$, then the roots of $Q$.

\end{enumerate}

\end{enumerate}

vague roostBOT
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Kuroo
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maiden dagger
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Without any Latex issues but in French .

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I need help for 4.c) please .

lusty yarrow
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There were some problems with Latex about the formulas so I sent the statement, so as to have the right formulas.

ember elmBOT
#

Chebychev's polynomial exercise problem

autumn sedge
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autumn sedge
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if indices confuse you then try to write it for some small n=5

maiden dagger
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Yes, I also did that. I also made changes to clues but I end up not finding what I should. If you want I can send what I did but I don't know if it will be really readable.

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