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A piecewise linear and
a polynomial interpolation.

The following theorem is called theorem about polynomial interpolation and describes the interpolation of given function values by a polynomial. If just one function value at one point is given, then this determines a constant polynomial, two values at two points determine a linear polynomial (the graph is a line), three values at three points determine a quadratic polynomial, etc.

Let be a field, and let different elements and elements are given. Then there exist a unique polynomial of degree , such that holds for all .

We prove the existence and consider first the situation where for all for some fixed . Then

is a polynomial of degree which at the points has value . The polynomial

has at these points still a zero, but additionally at its value is . We denote this polynomial by . Then

is the polynomial looked for, because for the point we have

for and .

The uniqueness follows from fact.

If the data and are given, then one can find the interpolating polynomial of degree , which exists by fact, in the following way: We write

with unknown coefficients , and determine then these coefficients. Each interpolation point yields a linear equation

over . The resulting system of linear equations has exactly one solution , which gives the polynomial.