Write $$p(x) = a \cdot \prod_{i=1}^{k} (x_i - z_i)$$with $x_i = z_i$ a complex solution or a real one, doesn't matter. Main claim is based on the following argument: $x_i - z_i | x_i^n - z_i^n$, therefore, there exists a polynomial $q_i(x)$ so that $(x_i-z_i) \cdot q_i(x) = x_i^n - z_i^n$, so that we get a new polynomial whose powers are all multiples of $n$. Now, do this for all $1 \le i \le k$, and we can get a polynomial that suffices by doing $$p(x) \cdot \prod_{i=1}^{k}q_i(x) = \prod_{i=1}^{k}(x_i^n-z_i^n)$$, giving us $k$ polynomials of degree $n$, meaning that all powers that are multiples of $n$ will have real coëfficient and others wont.