We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
115
1
avatar

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
 #1
avatar+6045 
+1

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

 

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

.
 Jul 30, 2019

5 Online Users

avatar