grandes-ecoles 2020 Q22

grandes-ecoles · France · centrale-maths2__psi Roots of polynomials Multiplicity and derivative analysis of roots

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 $k \in \llbracket 1, n \rrbracket$, $A_k'(X) = A_{k-1}(X - a)$.

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 $k \in \llbracket 1, n \rrbracket$, $A_k'(X) = A_{k-1}(X - a)$.

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