Interpolating polynomials If I interpolate a polynomial of degree n with n+1 points, will I always get the polynomial itself back? if so, does it work for k < n+1 points?

Suppose that $p,q$ are two polynomials of degree $n$ such that $p(x_i) = q(x_i)$ for $i=0, \dots ,n$. Then $p=q$.

In fact, if the polynomial $p-q \
eq 0$, then it has $n+1$ distinct zeroes, but it has degree less than $n$: this is impossible.

Note that this argument does not work for a smaller number of points.

Now, if you have a polynomial of degree $n$, when you interpolate it on $n+1$ points you get a polynomial of degree $n$ coinciding on your original polynomial in those points. This means that you get your original polynomial back.