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]] .](Q28368.png)
$${P}\:{is}\:{apolynomial}\:{from}\:{C}_{{n}} \left[{x}\right]\:{having}\:{n}\:{roots} \\ $$$$\left({x}_{{i}} \right)_{\mathrm{1}\leqslant{i}\leqslant{n}\:} \:\:\:\:{and}\:{x}_{{i}} #\:{x}_{{j}} \:{for}\:{i}#{j} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\:\sum_{{i}=\mathrm{1}} ^{{n}} \:\:\:\frac{\mathrm{1}}{{p}^{'} \left({x}_{{i}} \right)}\:\:=\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:\:\:\:\sum_{{i}=\mathrm{1}} ^{{n}} \:\:\:\frac{{x}_{{i}} ^{{k}} }{{p}^{'} \left({x}_{{i}} \right)}\:\:\:\:{with}\:{k}\in\left[\left[\mathrm{0},{n}−\mathrm{1}\right]\right]\:\:. \\ $$