# 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

Please help me develop my understanding of these methods more than just memorizing a formula.

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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
While learning this technique, keep in mind also that the technique is easily misused. Polynomial fits can often give wildly oscillatory curves that are not reasonable as interpolations. The higher the degree of the polynomial, the worse this can get. –  Ben Crowell Jul 31 '12 at 12:54
To add to @Ben's comments, there is what is called the "Runge phenomenon"; this is why polynomial interpolations of very high degree are not usually done. –  Ｊ. Ｍ. Jul 31 '12 at 13:19
Horner's algorithm is not an interpolation method, but a method for evaluating polynomials for a specified argument. If you're familiar with "synthetic division", this is related to the Horner scheme. –  Ｊ. Ｍ. Jul 31 '12 at 13:20
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