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

P is a polynomial havng n roots (x_i )_(1≤i≤n) with x_i ≠ x_j for i≠ j find the values of Σ_(k1) ^(k=n) (1/(x−x_k )) and Σ_(k=1) ^n (1/((x−x_k )^2 )) .

Pisapolynomialhavngnroots(xi)1⩽i⩽nwithxi≠xjfori≠jfindthevaluesof∑k1k=n1x−xkand∑k=1n1(x−xk)2.

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

P(x)=α Π_(k=1) ^(k=n) (x−x_k )⇒ ((P^ (x))/(P(x)))= Σ_(k=1) ^(k=n) (1/(x−x_k )) and after derivation ((P^′^′ (x)P(x)−(P^ )^2 )/((P(x))^2 ))= −Σ_(k=1) ^n (1/((x−x_k )^2 )) ⇒ Σ_(k=1) ^(k=n) (1/((x−x_k )^2 )) = ((((dP/dx))^2 − P(x)(d^2 P/dx^2 ))/((P(x))^2 )).

P(x)=α∏k=1k=n(x−xk)⇒P(x)P(x)=∑k=1k=n1x−xkandafterderivationP′′(x)P(x)−(P)2(P(x))2=−∑k=1n1(x−xk)2⇒∑k=1k=n1(x−xk)2=(dPdx)2−P(x)d2Pdx2(P(x))2.

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