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

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

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

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

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

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