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Find the sum of the roots, real and non-real, of the equation \(x^{2001}+\left(\frac 12-x\right)^{2001}=0\), given that there are no multiple roots.

 Jul 30, 2019
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\(x^{2001}+\left(\dfrac 1 2 - x \right)^{2001} = \\ x^{2001} -x^{2001} + \sum \limits_{k=1}^{2001}\left(\dfrac 1 2\right)^k (-x)^{2001-k} = \\ \sum \limits_{k=1}^{2001}\left(\dfrac 1 2\right)^k (-x)^{2001-k} = \\ \dfrac 1 2 \sum \limits_{k=0}^{2000}\left(\dfrac 1 2\right)^k (-x)^{2000-k} = \\ \dfrac 1 2\left(\dfrac 1 2 - x\right)^{2000} = 0\)

 

\(\text{$x=\dfrac 1 2$ is the only solution and it is a root of order 2000}\)

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 Jul 30, 2019

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