Polynomial/K/Interpolation/Fact/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.