Another way to look at this problem is to set up a system of linear congruences. We can look at a concrete example first - you should be able to generalize it from there.

Suppose we want to find a polynomial such that . If we treat as an unknown, we have a system of linear (polynomial) congruences:

If you run the Chinese Remainder Theorem algorithm on this, you find that the polynomial is . It is automatically guaranteed to be unique by the statement of the theorem.