P-is-a-polynomial-havng-n-roots-x-i-1-i-n-with-x-i-x-j-for-i-j-find-the-values-of-k1-k-n-1-x-x-k-and-k-1-n-1-x-x-k-2-

Question Number 27000 by abdo imad last updated on 01/Jan/18

P i s a p o l y n o m i a l h a v n g n r o o t s {(x_{i})}_{1 ⩽ i ⩽ n} w i t h x_{i} \neq x_{j} f o r i \neq j

f i n d t h e v a l u e s o f \sum_{k 1}^{k = n} \frac{1}{x - x_{k}} a n d \sum_{k = 1}^{n} \frac{1}{{(x - x_{k})}^{2}} .

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

P (x) = α \prod_{k = 1}^{k = n} (x - x_{k}) \Rightarrow \frac{P^{} (x)}{P (x)} = \sum_{k = 1}^{k = n} \frac{1}{x - x_{k}} a n d a f t e r

d e r i v a t i o n \frac{P^{^{'^{'}}} (x) P (x) - {(P^{})}^{2}}{{(P (x))}^{2}} = - \sum_{k = 1}^{n} \frac{1}{{(x - x_{k})}^{2}}

\Rightarrow \sum_{k = 1}^{k = n} \frac{1}{{(x - x_{k})}^{2}} = \frac{{(\frac{d P}{d x})}^{2} - P (x) \frac{d^{2} P}{{d x}^{2}}}{{(P (x))}^{2}} .