Polynomial/K/Interpolation/System of linear equations/Remark

If the data ${}a_{1},\ldots ,a_{n}$ and ${}b_{1},\ldots ,b_{n}$ are given, then one can find the interpolating polynomial ${}P$ of degree ${}\leq n-1$, which exists by fact, in the following way: We write

${}P=c_{0}+c_{1}X+c_{2}X^{2}+\cdots +c_{n-2}X^{n-2}+c_{n-1}X^{n-1}\,$

with unknown coefficients ${}c_{0},\ldots ,c_{n-1}$, and determine then these coefficients. Each interpolation point ${}(a_{i},b_{i})$ yields a linear equation

${}c_{0}+c_{1}a_{i}+c_{2}a_{i}^{2}+\cdots +c_{n-2}a_{i}^{n-2}+c_{n-1}a_{i}^{n-1}=b_{i}\,$

over ${}K$. The resulting system of linear equations has exactly one solution ${}(c_{0},\ldots ,c_{n-1})$, which gives the polynomial.