grandes-ecoles 2025 Q1

grandes-ecoles · France · mines-ponts-maths1__pc Roots of polynomials Reciprocal and antireciprocal polynomial properties

Let $P \in \mathbf{C}[X]$ of degree $p$. We write $P = \sum_{k=0}^{p} a_k X^k$, where $a_0, \ldots, a_p$ are complex numbers, and $a_p \neq 0$.
Show that $P$ is reciprocal if and only if for every integer $k$, $0 \leq k \leq p$, we have the equality $a_k = a_{p-k}$.

Let $P \in \mathbf{C}[X]$ of degree $p$. We write $P = \sum_{k=0}^{p} a_k X^k$, where $a_0, \ldots, a_p$ are complex numbers, and $a_p \neq 0$.

Show that $P$ is reciprocal if and only if for every integer $k$, $0 \leq k \leq p$, we have the equality $a_k = a_{p-k}$.

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