Two orthogonal basis of polynomials with respect to a same inner product have the same roots
Solution 1:
With little loss we can assume the $p_k$ have been arranged so that they have unit norm, namely $\langle p_k, p_k \rangle = 1$.
You have defined $c_n$ such that $q_n = \sum_k c_k p_k$. Then for $k < n$, using orthogonality of the $p_k$, $$\langle q_n, p_k\rangle = c_k.$$ But the left side is also zero since $q_n$ is orthogonal to every polynomial of lower degree. Thus $c_0, c_1, c_{n-1} = 0$ and $q_n$ is a multiple of $p_n$ and both must have the same roots.