I am currently stuck with the following question:

Let $q$ be a polynomial of degree $n+1$ with distinct positive zeros $x_0, ... , x_n$. Find a polynomial $p \in P_n$ that satisfies the functional equation $q(x)p(x) + q(-x)p(-x) = 1$ for all $x$ in $\Re$. Is such a polynomial unique?

Would appreciate your kind explanations! Thank you!