Reducibility of polynomial in rational field

Question

Let $f(x)\in \mathbb{Q}[x]$ be a polynomial of degree $n>0$. Let $p_1, p_2, \dots ,p_{n+1}$ be distinct prime numbers. Show that there exists a non-zero polynomial $g(x)\in \mathbb{Q}[x]$ such that $fg=\sum_{i=1}^{n+1} c_ix^{p_i}$ with every $c_i\in \mathbb{Q}$.