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### Fundamental theorem of mathematics

The **fundamental theorem of mathematics** says that you can always make a function in the form

##### Explanation

It is always possible to define a function, which connects any number of points, which lie in a flat plane. For two points this might be a straight line, but it could also be a hyperbola. Three points may still be on a straight line, but then these points must be on the same line.

The fundamental theorem of mathematics describes this. In short

With this formula you can determine functions by extrapolation.

##### Example 1

The function *f* (*x*) = (1 + *x*)^{n} can be written as

and for *x* = 0 it gives 1 = *a*_{0}. The first derivative is

and for *x* = 0 it gives *n* = *a*_{1}. The second derivative is

and for *x* = 0 it gives *n*(*n* − 1) = 2*a*_{2}. By substitution of *a*_{0}, *a*_{1} and *a*_{2} you get

This is a special version of the binomium of Newton.