# Find the sum of the roots given no multiple roots.

Find the sum of the roots, real and non-real, of the equation $$x^{2001} + \left( \frac{1}{2} - x \right)^{2001} = 0$$ given that there are no multiple roots.

I am in a weird situation here.

$$x^{2001} = -\left( \frac{1}{2} - x \right)^{2001}$$

$$x = -\frac{1}{2} + x$$

Then I get $0 = -\frac{1}{2}$ which I know is not true.

So all roots have to be complex roots. By vieta's formulas:

$$\sum _{k=1}^{n} r_k = -\frac{a_{n-1}}{a_n}$$

But that cant be of help here. The leading coefficient is $Cx^{2000}$.