Let $n$ be a positive integer,and let $f(x)$ be a function defined on a domain containing the $n+1$ distinct points $x_0,x_1,\cdots,x_n$,and let $p_n(x)$ be the polynomial of degree $n$ that interpolates $f(x)$ at the points $x_0,x_1,\cdots,x_n$.For each $i=0,1,\cdots,n$,we define $p_{n-1,i}(x)$ to be the polynomial of degree $n-1$ that interpolates $f(x)$ at the points $x_0,x_1,\cdots,x_{i-1},x_{i+1},\cdots,x_n$.If $i$ and $j$ are distinct nonnegative integers not exceeding $n$,then
\begin{equation}
p_n(x)=\frac{(x-x_j)p_{n-1,j}(x)-(x-x_i)p_{n-1,i}(x)}{x_i-x_j}
\end{equation}
Proof:This is very easy.$p_n(x)$ and $\frac{(x-x_j)p_{n-1,j}(x)-(x-x_i)p_{n-1,i}(x)}{x_i-x_j}$ are two polynomials of the degree $n$.At each point of $x_0,x_1,\cdots,x_n$ the output of the two polynomials are equal,so the two polynomials are equal.
注:两点确定一条直线,三点确定一条二次函数,四点确定一条三次函数……多项式函数带给我一种强烈的确定感.它太确定了，让我总觉得自己对它的认识还不够.这道题目的结论无疑让我更明白了一层.