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

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

\frac{b_{i}}{(a_{i} - a_{1}) \dots (a_{i} - a_{i - 1}) (a_{i} - a_{i + 1}) \dots (a_{i} - a_{n})} (X - a_{1}) \dots (X - a_{i - 1}) (X - a_{i + 1}) \dots (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} + \dots + 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.