grandes-ecoles 2025 Q3

grandes-ecoles · France · mines-ponts-maths1__psi Roots of polynomials Reciprocal and antireciprocal polynomial properties

Let $Q$ be a polynomial of degree $p$. We say that $Q$ is antireciprocal if $$Q(X) = -X^p Q\left(\frac{1}{X}\right)$$ Show that if $Q$ is antireciprocal, 1 is a root of $Q$ and that there exists a polynomial $P$ that is constant or reciprocal such that $Q = (X-1)P$.

Let $Q$ be a polynomial of degree $p$. We say that $Q$ is antireciprocal if
$$Q(X) = -X^p Q\left(\frac{1}{X}\right)$$
Show that if $Q$ is antireciprocal, 1 is a root of $Q$ and that there exists a polynomial $P$ that is constant or reciprocal such that $Q = (X-1)P$.

Paper Questions

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