Here goes: remember the statement "two points determine a line"? The generalization of this statement to polynomials is that "$n+1$ conditions are needed to uniquely determine an $n$-th degree polynomial". That is, "three points determine a quadratic", "four points determine a cubic", ... you get the drift.
You were given six points: thus, any uniquely determined polynomial that must pass through those six points (you are interpolating, after all) should have a degree at most five (there are polynomials with degree $> 5$ that pass through those points, but you won't be able to pin them down unless you've other conditions). It can happen that the degree of the underlying interpolating polynomial might be less, but there's a way to check for that: take successive differences repeatedly until you encounter a constant sequence ($2.9-0, 14.9-2.8, \dots$ and then keep repeating). When I try it out, I've needed five iterations to reach a constant sequence, so an interpolating polynomial of degree $5$ should be expected.
However, there is a related procedure, called fitting or regression, that attempts to derive a polynomial of degree usually less than $n$ that is a "best approximation" of your data. Such a function won't pass through the given points, however, and the underlying assumptions are different.