grandes-ecoles 2020 Q21

grandes-ecoles · France · centrale-maths2__psi Roots of polynomials Polynomial evaluation, interpolation, and remainder

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 $$A_0 = 1 \quad \text{and, for all } k \in \llbracket 1, n \rrbracket, \quad A_k = \frac{1}{k!} X(X - ka)^{k-1}.$$ We denote by $\mathbb{C}_n[X]$ the $\mathbb{C}$-vector space of polynomials with complex coefficients and degree at most $n$. Prove that the family $(A_0, \ldots, A_n)$ is a basis of $\mathbb{C}_n[X]$.

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
$$A_0 = 1 \quad \text{and, for all } k \in \llbracket 1, n \rrbracket, \quad A_k = \frac{1}{k!} X(X - ka)^{k-1}.$$
We denote by $\mathbb{C}_n[X]$ the $\mathbb{C}$-vector space of polynomials with complex coefficients and degree at most $n$. Prove that the family $(A_0, \ldots, A_n)$ is a basis of $\mathbb{C}_n[X]$.

Paper Questions

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