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The Gauss Lucas Theorem

We prove the Gauss-Lucas Theorem. If we are given a polynomial with complex coefficients, then we consider the roots of the polynomial and the roots of the derivative of the polynomial. We use logarithmic differentiation to prove that. the roots of the derivative lie in the convex hull of the roots of the polynomial. We also state Marden's Theorem (which we will prove later) which states that the roots of the derivative of a cubic polynomials are the foci of an inellipse of the triangle formed by the roots of the polynomial. In the definition of convex hull, the condition that the sum t_1+ ... + t_n = 1 is missing! Please add that in your notes! #mikethemathematician, #mikedabkowski, #profdabkowski, #complexanalysis, #gauss-lucas

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