TITLE: Prove $\sum_{i=0}^{n}\left(x_{i}^{n}\prod_{0\leq k\leq n}^{k\neq i}\frac{x-x_k}{x_i-x_k}\right)=x^n$
QUESTION [7 upvotes]: Suppose $x_0$ , $x_1$ , $x_2$ , ... , $x_n$ are distinct real numbers , prove that :
$$
\large{\displaystyle{\sum_{i=0}^{n}
\left(
x_{i}^{n}\prod_{\substack{0\leq k\leq n \\ k\neq i }}\frac{x-x_k}{x_i-x_k}
\right)=x^n}}
$$
I have no ideas to do this question

REPLY [6 votes]: That's simply Lagrange's polynomial interpolation formula for the values of the polynomial $x^n$. Since there are $n+1$ data points, the two polynomials coincide.