Generating a polynomial for interpolation

Question

I'm looking into different types of interpolation, I was wondering how to generate a polynomial based on given data points when looking to interpolate. Looking on the wiki it tends to say things along the line of "suppose the polynomial is in the form ....", how do they arrive at this polynomial that represents the nature of the data set?

Answer 1

For a polynomial which passes through all given points exactly, see Lagrange polynomial.

For a polynomial of given degree which best approximates your points, see Polynomial regression.

Aside from the degree, neither of these techniques makes any assumptions about the form of the polynomial.