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
-
![{\displaystyle {}P=c_{0}+c_{1}X+c_{2}X^{2}+\cdots +c_{n-2}X^{n-2}+c_{n-1}X^{n-1}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/447f0fc870ced1fc106906adbd6b8873941f43e9)
with unknown coefficients
, and determine then these coefficients. Each interpolation point
yields a linear equation
-
![{\displaystyle {}c_{0}+c_{1}a_{i}+c_{2}a_{i}^{2}+\cdots +c_{n-2}a_{i}^{n-2}+c_{n-1}a_{i}^{n-1}=b_{i}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a326578b0c33bfbb745f63a798e06abd5cbbbdb6)
over
. The resulting system of linear equations has exactly one solution
, which gives the polynomial.