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How do I solve the equation given below for values for $\epsilon$?

$ \sum_{j=1}^{n-1} (2{\alpha_{j}}+1) ~~ \prod_{k=1,~k\neq j}^{n-1} \bigg(({\alpha_{k}}-1){\Big((2{\alpha_{k}}+3) + {\epsilon}(2{\alpha_{k}}+1)\Big)}^{2}\bigg) = 0 $

where $\alpha_i = \cos{\dfrac{2\pi i}{n}}$, where $n$ is an even integer and $n \geqslant 8$. ($j$ is NOT the imaginary unit).

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    @ GEdgar - I haven't tried when $n$ is 2 or 3 since my problem is basically to find the behavior when $n$ is greater than or equal to 8.2011-11-01

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