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Question Number 104775 by MAB last updated on 23/Jul/20

let P_n  an n-polynomial.  let a_1 ,...,a_n  its simple roots   let m_k  the slope of the tangent to P_n  at  the point (a_k ,0)  prove that  Σ_(k=1) ^n (1/m_k )=0  what about multiple roots?

$${let}\:{P}_{{n}} \:{an}\:{n}-{polynomial}. \\ $$$${let}\:{a}_{\mathrm{1}} ,...,{a}_{{n}} \:{its}\:{simple}\:{roots}\: \\ $$$${let}\:{m}_{{k}} \:{the}\:{slope}\:{of}\:{the}\:{tangent}\:{to}\:{P}_{{n}} \:{at} \\ $$$${the}\:{point}\:\left({a}_{{k}} ,\mathrm{0}\right) \\ $$$${prove}\:{that} \\ $$$$\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{m}_{{k}} }=\mathrm{0} \\ $$$${what}\:{about}\:{multiple}\:{roots}? \\ $$$$ \\ $$

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