There is a slight part of the following proof in my textbook which I don't quite get.
If $x_0, x_1,...,x_n$ are distinct real numbers, then for arbitrary values $y_0, y_1,...,y_n$, there is a unique polynomial $p_n$ of degree at most $n$ such that
$$p_n(x_i) = y_i \quad (0 \leq i \leq n)$$
Let us prove the uniqueness or unicity first. Suppose there were two such polynomials, $p_n$ and $q_n$. Then the polynomial $p_n - q_n$ would have the property $(p_n - q_n)(x_i) = 0$ for $0 \leq i \leq n$. Since the degree of $p_n - q_n$ can be at most $n$, this polynomial can have at most $n$ zeros if it is not the $0$ polynomial. Since the $x_i$ are distinct, $p_n - q_n$ has $n+1$ zeros; it must therefore be $0$. Hence, $p_n = q_n$.
OK, so the only thing I don't quite here is the part:
"Since the $x_i$ are distinct, $p_n - q_n$ has $n+1$ zeros"
How does it follow that this must have exactly $n+1$ zeros. I have pondered this for some time, but just can't see why this is obvious. I'm probably overlooking something simple here, but if anyone can help me out, I would greatly appreciate it!