Discussion: This student uses a representation of the polynomial to understand what the condition means, essentially that the k th coefficient and the n+1-k th coefficient are negatives of one another (and therefore if n is odd then there is n+1/2 coefficient is zero. Then the polynomial is the sum of polynomials of the form ax^k(x^i-1)$ and these are each divisible by x-1. This could be considered a brute force approach. By evaluating at 1 we fiind that f(1) = -f(1) and so f(1) = 0 and therefore by the root-factor theorem x-1 divides f(x).

Written comment: Nice treatment of this. Have you thought about what f(1) is equal to and what consequences you can draw from this?