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  "body": "\n## Statement\n\nLet $P$ be a nonconstant polynomial with coefficients in the [[complex numbers|09FTAET5Z4]], and let $P'$ be its derivative.\nThen every [[root|00EHX6FWC0]] of $P'$ lies in the [[convex hull|0TCHAS04WK]] of the set of roots of $P$, viewed as points of the plane $\\C$.\n\n## Remarks\n\nProof idea: write $P(z) = c \\prod_{i=1}^{n} (z - z_i)$, so that $\\frac{P'(z)}{P(z)} = \\sum_{i=1}^{n} \\frac{1}{z - z_i}$ when $P(z) \\ne 0$.\nLet $w$ be a root of $P'$; if $P(w) = 0$ there is nothing to prove.\nOtherwise $\\sum_i \\frac{1}{w - z_i} = 0$; taking complex conjugates gives $\\sum_i \\frac{w - z_i}{\\abs{w - z_i}^2} = 0$.\nSo $w = \\sum_i \\lambda_i z_i$ with $\\lambda_i = \\abs{w - z_i}^{-2} \\big/ \\sum_j \\abs{w - z_j}^{-2}$, a convex combination of the $z_i$.\n\nExample: $P(z) = z^3 - 1$ has roots the three cube roots of unity, and $P'(z) = 3z^2$ has the root $0$, the centre of the triangle they form.\n\nTake a real polynomial whose roots are all real.\nThen the theorem says that the roots of $P'$ lie between the smallest and the largest root of $P$, which also follows from [[Rolle's theorem|0194DY35BG]].\n\n## Explanations\n\n### L1 human\n\nMark the roots of a polynomial as points in the plane and stretch a rubber band around them.\nThe roots of the derivative all lie inside the band.\nDifferentiating never moves the roots outside the region spanned by the original ones.\n\n### L2 human\n\nThe theorem is a complex analogue of Rolle's theorem.\nApplied repeatedly, it shows that the roots of all derivatives $P^{(k)}$, $k \u003c \\deg P$, lie in the convex hull of the roots of $P$.\nIt is a basic tool in the geometry of polynomials and is used, for example, in a short proof of [[Bernstein's inequality for polynomials|F0YGC89S9R]].\n",
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