This is called Lagrange interpolation.
Let be the polynomial which satisfies for . Then you have
The values are known, and this is a linear system of equations in the unknown parameters . Solving for those will give you the polynomial you seek.
Note that the matrix of coefficients of the system is a Vandermonde matrix, whose inverse is known explicitly. I suspect that with a bit of cleverness, the inverse of the Vandermonde matrix may be retreived from Lagrange's form of the interpolating polynomial.