Question Number 34607 by abdo mathsup 649 cc last updated on 08/May/18
![let p∈C[x] degp=n (x_i )_(1≤k≤n) the roots of p(x) a∈C?/p(a)≠0 1) calculate S_1 = Σ_(k=1) ^n (1/(x_k −a)) interms of p,p^′ and a 2)calculste S_2 =Σ_(k=1) ^n (1/((x_k −a)^2 )) interms of p,p^, p^(′′) and a.](Q34607.png)
$${let}\:{p}\in{C}\left[{x}\right]\:{degp}={n}\:\:\:\left({x}_{{i}} \right)_{\mathrm{1}\leqslant{k}\leqslant{n}} {the}\:{roots}\:{of}\:{p}\left({x}\right) \\ $$$${a}\in{C}?/{p}\left({a}\right)\neq\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{S}_{\mathrm{1}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{x}_{{k}} −{a}}\:{interms}\:{of}\:{p},{p}^{'} \:{and}\:{a} \\ $$$$\left.\mathrm{2}\right){calculste}\:{S}_{\mathrm{2}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{\left({x}_{{k}} −{a}\right)^{\mathrm{2}} }\:\:{interms}\:{of}\:{p},{p}^{,} \\ $$$${p}^{''} \:{and}\:{a}. \\ $$