let-P-n-an-n-polynomial-let-a-1-a-n-its-simple-roots-let-m-k-the-slope-of-the-tangent-to-P-n-at-the-point-a-k-0-prove-that-k-1-n-1-m-k-0-what-about-multiple-roots-

Question Number 104775 by MAB last updated on 23/Jul/20

l e t P_{n} a n n - p o l y n o m i a l .

l e t a_{1}, \dots, a_{n} i t s s i m p l e r o o t s

l e t m_{k} t h e s l o p e o f t h e t a n g e n t t o P_{n} a t

t h e p o i n t (a_{k}, 0)

p r o v e t h a t

\underset{k = 1}{\sum^{n}} \frac{1}{m_{k}} = 0

w h a t a b o u t m u l t i p l e r o o t s ?