Let Q(z)=(z−α_1)⋯(z−α_n) be a polynomial of degree >1 with distinct roots outside the real line. sum_(j=1)^n /(1)(Q'(alpha_j))=0. Do we have a proof relying on rudimentary techniques?

Medenovgj 2022-09-26 Answered
Let Q ( z ) = ( z α 1 ) ( z α n ) be a polynomial of degree > 1 with distinct roots outside the real line.
We have
j = 1 n 1 Q ( α j ) = 0.
Do we have a proof relying on rudimentary techniques?
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Answers (1)

Micah Hobbs
Answered 2022-09-27 Author has 8 answers
Let n 1 and α 1 , , α n C be distinct. If we put Q ( z ) = ( z α 1 ) ( z α n ), then from the partial fraction decomposition, it follows that
1 Q ( z ) = j = 1 n 1 Q ( α j ) ( z α j ) .
From this, we have
j = 1 n 1 Q ( α j ) = lim | z | j = 1 n z Q ( α j ) ( z α j ) = lim | z | z Q ( z ) = { 1 , if  n = 1 0 , if  n > 1 .
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