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If \( \alpha_{1}, \alpha_{2}, \ldots, \alpha_{m} \) are the roots of polynomial equation
\[
f(x)=a_{0} x^{m}+a_{1} x^{m-1}+\ldots+a_{m-1} x+a_{m}=0,
\]
then
\[
f(x)=a_{0}\left(x-\alpha_{1}\right) \ldots\left(x-\alpha_{m}\right) .
\]
and \( \quad \frac{f^{\prime}(x)}{f(x)}=\frac{1}{x-\alpha_{1}}+\cdots+\frac{1}{x-\alpha_{m}} \)
If \( \omega=\cos \left(\frac{2 \pi}{n}\right)+i \sin \left(\frac{2 \pi}{n}\right) \), then numerical value
\[
1+\frac{1}{2-\omega}+\frac{1}{2-\omega^{2}}+\cdots+\frac{1}{2-\omega^{n-1}}
\]
equals
(a) \( \frac{n\left(2^{n-1}\right)}{2^{n}+1} \)
(b) \( \frac{n\left(2^{n-1}\right)}{2^{n}-1} \)
(c) 0
(d) 1
\( \mathrm{P} \)
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