There exists a unique polynomial of degree $n$ passing through $(n+1)$ points in a plane.

So, if one we have a one-point say $(a,b)$ then clearly $f(x)=b$ is the (unique) polynomial of degree $0=1-1$ through the point. Now, we all know, $\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}$ is the unique polynomial (put $y=f(x))$ of degree 1.

Hence we sort of asking the general question and fortunately, the answer is yes, which one proves using the Vandermonde matrix and of course Lagrange Interpolation formula actually gives the polynomial.

Here is the solution,

Reference:
https://math.stackexchange.com/questions/837902/prove-there-exists-a-unique-n-th-degree-polynomial-that-passes-through-n1-p


Inverse of a Vandermonde Matrix : ​https://proofwiki.org/wiki/Inverse_of_Vandermonde_Matrix