# Should n points always be interpolated by n+1 degree polynomial?

I'm studying interpolation and I see that if you have 2 points you use a 3rd-degree polynomial and likewise a 6th degree polynomial for five points. Is this a general formula, and if so, what is it called( and how to prove it)?

These are the interpolation techniques I'm looking to learn:

• Naive interpolation
• Newtonian interpolation
• Lagrange interpolation
• Hermite interpolation Horner's algorithm

No, two points determine a line, which has degree $1$. In any event, look up Lagrange interpolation. –  Ｊ. Ｍ. Jul 31 '12 at 12:38
Hermite interpolation is the form of polynomial interpolation that takes into account derivatives (slopes), in addition to function values. In general, you need $n+1$ conditions to uniquely determine a polynomial of degree $n$ (to be able to set up the linear system for solving fir the coefficients). So, a cubic (degree 3) can be determined by four points (Lagrange), or two points and two derivatives (Hermite). –  Ｊ. Ｍ. Jul 31 '12 at 12:49