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Question Number 32334 by abdo imad last updated on 23/Mar/18

$$pisapolynomialhavingnroots\left(x_i\right)with{x}_i\ne xjfor$$$$i\ne jcalculate\sum _{k=1}^n\frac{1}{1-x_k}.$$

Commented by abdo imad last updated on 01/Apr/18

$$p\left(x\right)=\lambda \prod _{k=1}^n(x-x_i)\Rightarrow p^{{}^{\prime }}\left(x\right)=\lambda \sum _{k=1}^n\prod _{p=1_{p\ne k}}^n(x-x_p)$$$$\Rightarrow \frac{p^{{}^{\prime }}\left(x\right)}{p\left(x\right)}=\sum _{k=1}^n\frac{1}{x-x_k}\Rightarrow \sum _{k=1}^n\frac{1}{1-x_k}=\frac{p^{{}^{\prime }}\left(1\right)}{p\left(1\right)}withp\left(1\right)\ne 0$$$$$$

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