Loterre: Mathématiques: Gauss-Lucas theorem

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Gauss-Lucas theorem

In complex analysis, a branch of mathematics, the Gauss–Lucas theorem gives a geometric relation between the roots of a polynomial P and the roots of its derivative $P'$ . The set of roots of a real or complex polynomial is a set of points in the complex plane. The theorem states that the roots of $P'$ all lie within the convex hull of the roots of P, that is the smallest convex polygon containing the roots of P. When P has a single root then this convex hull is a single point and when the roots lie on a line then the convex hull is a segment of this line. The Gauss–Lucas theorem, named after Carl Friedrich Gauss and Félix Lucas, is similar in spirit to Rolle's theorem.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Gauss%E2%80%93Lucas_theorem)

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