Polynomial/K/Interpolation/Fact/Proof

Proof

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

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

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

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

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

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

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

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

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

The uniqueness follows from fact.

To fact