Then you can use elimination to solve $Ax = b$, as I've done here.

Efficiency

Elimination is $O(n^3)$, Realistically you should use $O(n^2)$ with Lagrange polynomials wiki, video. (also related to how $A$ is a Vandermonde matrix).

But I like the generality of elimination :D

There are also much more sophisticated techniques, like the Fast Fourier transform which (I think?) is $O(n \log n)$. I don't understand it yet though.