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-k-1-n-1-m-k-0-what-about-multiple-roots-

Tinku Tara June 4, 2023 None 0 Comments

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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