if a_1 ,a_2 ,...a_(14 ) are roots of the polynomial p(x)=x^(14) +x^8 2x+1 calculate Σ_(i=1) ^(14) (1/((a


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Question Number 28544 by abdo imad last updated on 26/Jan/18

$i f a_{1}, a_{2}, . . . a_{14} a r e r o o t s o f t h e p o l y n o m i a l$ $p (x) = x^{14} + x^{8} 2 x + 1 c a l c u l a t e \sum_{i = 1}^{14} \frac{1}{{(a_{i} - 1)}^{2}} .$
Commented by abdo imad last updated on 26/Jan/18

$p (x) = x^{14} + x^{8} + 2 x + 1 .$
Commented by abdo imad last updated on 28/Jan/18

$w e k n o w t h a t \frac{p^{^{'}} (x)}{p (x)} = \sum_{i = 1}^{14} \frac{1}{x - x_{i}}$ $\frac{p^{^{″}} p (x) - p^{' 2} (x)}{p^{2} (x)} = - \sum_{i = 1}^{14} \frac{1}{{(x - x_{i})}^{2}} b u t w e h a v e$ $p^{^{'}} (x) = 14 x^{13} + 8 x^{7} + 2 a n d p^{(2)} (x) = 14 \times 13 x^{12} + 56 x^{6}$ $\sum_{i = 2}^{14} \frac{1}{(x - a_{i}) 2} = - \frac{(14 \times 13 x^{12} + 56 x^{6}) (x^{14} + x^{8} + 2 x + 1) - {(14 x^{13} + 8 x^{7} + 2)}^{2}}{{(x^{14} + x^{8} + 2 x + 1)}^{2}}$ $a n d f o r x = 1 ? w e f i n d$ $\sum_{i = 1}^{14} \frac{1}{{(a_{i} - 1)}^{2}} = - \frac{(14 \times 13 + 56) \times 5 - (24)}{25} . . .$
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