You say this is a polynomial. It cannot be a polynomial in c or n, since c and n live in the exponentis of p. Therefore you are telling us it is a polynomial in p.
It ofen helps to get some feeling for what is happening if we try some values for the parameters. So I picked some values for c and n.
- 324*p^11 + 1782*p^10 - 4620*p^9 + 7425*p^8 - 7920*p^7 + 5544*p^6 - 2376*p^5 + 495*p^4 - 6*p
As I said, the result is a polynomial in p.
But then it makes no sense at all to tell us that p, 1-p, and 2*p-1 are "roots". They may be FACTORS of this polynomial. The corresponding roots would be 0, 1, and 1/2. Is that true in this case?
- 0.12634503327930217170060586438812 + 0.13836497949209092930010607856289i
1.1263450332793021717006058643881 - 0.13836497949209092930010607856289i
- 0.12634503327930217170060586438812 - 0.13836497949209092930010607856289i
1.1263450332793021717006058643881 + 0.13836497949209092930010607856289i
0.5 - 0.56660830776358934972132646114141i
0.5 + 0.56660830776358934972132646114141i
0.5 - 1.0883660924340429282641590517296i
0.5 + 1.0883660924340429282641590517296i
Anyway, your claim seems to hold, at least in this case.
A problem is however, if c and n are allowed to be completely unknown integers, is that symsum will not produce a result, beyond the call to symsum.
It should be not that difficult to show that 0, 1/2, and 1 are roots of the result. For example as long as there is no constant term in the polynomial, then p==0 is trivially a root. If the sum of the coefficients is equal to 0, then 1 would be a root.
However, I see no simple solution to show no other real root can exist for fully general integer values of c and n.
For example, since the coefficients themselves will be difficult to generate for fully general values of c nd n, you might have some difficulty in applying a simple tool like Descarte's rule of signs.
Perhaps your best approach is to try induction in some form. It looks like when c == 0, the polynomial is always trivially 0 for any n. But if you start with some lower case, for c and n, can you then write a recurrence relation that connects P(n,c), for example, and p(n+1,c)?
Now you may be able to prove something abut the roots of the family of polynomials, as n or c grows.