# Polynomial/K/Interpolation/Section

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.

## Theorem

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}n}$ different elements ${\displaystyle {}a_{1},\ldots ,a_{n}\in K}$ and ${\displaystyle {}n}$ elements ${\displaystyle {}b_{1},\ldots ,b_{n}\in K}$ are given. Then there exist a unique polynomial ${\displaystyle {}P\in K[X]}$ of degree ${\displaystyle {}\leq n-1}$, such that ${\displaystyle {}P{\left(a_{i}\right)}=b_{i}}$ holds for all ${\displaystyle {}i}$.

### Proof

We prove the existence and consider first the situation where ${\displaystyle {}b_{j}=0}$ for all ${\displaystyle {}j\neq i}$ for some fixed ${\displaystyle {}i}$. Then

${\displaystyle (X-a_{1})\cdots (X-a_{i-1})(X-a_{i+1})\cdots (X-a_{n})}$

is a polynomial of degree ${\displaystyle {}n-1}$ which at the points ${\displaystyle {}a_{1},\ldots ,a_{i-1},a_{i+1},\ldots ,a_{n}}$ has value ${\displaystyle {}0}$. The polynomial

${\displaystyle {\frac {b_{i}}{(a_{i}-a_{1})\cdots (a_{i}-a_{i-1})(a_{i}-a_{i+1})\cdots (a_{i}-a_{n})}}(X-a_{1})\cdots (X-a_{i-1})(X-a_{i+1})\cdots (X-a_{n})}$

has at these points still a zero, but additionally at ${\displaystyle {}a_{i}}$ its value is ${\displaystyle {}b_{i}}$. We denote this polynomial by ${\displaystyle {}P_{i}}$. Then

${\displaystyle {}P=P_{1}+P_{2}+\cdots +P_{n}\,}$

is the polynomial looked for, because for the point ${\displaystyle {}a_{i}}$ we have

${\displaystyle {}P_{j}(a_{i})=0\,}$

for ${\displaystyle {}j\neq i}$ and ${\displaystyle {}P_{i}(a_{i})=b_{i}}$.

The uniqueness follows from fact.

${\displaystyle \Box }$

## 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.