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P-is-apolynomial-from-C-n-x-having-n-roots-x-i-1-i-n-and-x-i-x-j-for-i-j-1-prove-that-i-1-n-1-p-x-i-0-2-find-i-1-n-x-i-k-p-x-i-with-k-




Question Number 28368 by abdo imad last updated on 24/Jan/18
P is apolynomial from C_n [x] having n roots  (x_i )_(1≤i≤n )     and x_i # x_j  for i#j  1) prove that   Σ_(i=1) ^n    (1/(p^′ (x_i )))  =0  2) find     Σ_(i=1) ^n    (x_i ^k /(p^′ (x_i )))    with k∈[[0,n−1]]  .
$${P}\:{is}\:{apolynomial}\:{from}\:{C}_{{n}} \left[{x}\right]\:{having}\:{n}\:{roots} \\ $$$$\left({x}_{{i}} \right)_{\mathrm{1}\leqslant{i}\leqslant{n}\:} \:\:\:\:{and}\:{x}_{{i}} #\:{x}_{{j}} \:{for}\:{i}#{j} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\:\sum_{{i}=\mathrm{1}} ^{{n}} \:\:\:\frac{\mathrm{1}}{{p}^{'} \left({x}_{{i}} \right)}\:\:=\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:\:\:\:\sum_{{i}=\mathrm{1}} ^{{n}} \:\:\:\frac{{x}_{{i}} ^{{k}} }{{p}^{'} \left({x}_{{i}} \right)}\:\:\:\:{with}\:{k}\in\left[\left[\mathrm{0},{n}−\mathrm{1}\right]\right]\:\:. \\ $$

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