VC dimension of the class of polynomial classifiers of degree $n$
The idea is that a polynomial of degree $n$ has at most $n$ roots, and so can change signs at most $n$ times. Therefore no polynomial of degree $n$ can form an alternating pattern +-+-... or -+-+... of length $n+2$. This shows that the VC dimension is at most $n+1$.

On the other hand, for any set of $n+1$ pairs $(x_1,y_1),\ldots,(x_{n+1},y_{n+1})$, there is a polynomial of degree $n$ which interpolates them, given by the Lagrange interpolation formula. Using $y_i = \pm 1$, you can easily show that any set of $n+1$ points is shattered. Therefore the VC dimension is exactly $n+1$.