Plugging into results in the sum of all roots of unity. This always equals for all .
Your original question though was to show is a root of . This is easy to see since .
Let and where p is a prime > 2.
Show that is a root of f(x). done...
Show that is a factor of f(x). done...
I used geometric series to extrapolate f(x)
Now, I am having trouble showing that is a root of g(x) for . When I just try to plug and play in g(x), I don't get anything that heads towards 0. Step by step or a skeleton is fine with me, I just need a bit of a push.