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

Suppose $x_0$ , $x_1$ , $x_2$ , ... , $x_n$ are distinct real numbers , prove that :

$$\large{\displaystyle{\sum_{i=0}^{n}\left(x_{i}^{n}\prod_{0\leq k\leq n}^{k\neq i}\frac{x-x_k}{x_i-x_k}\right)=x^n}}$$

I have no ideas to do this question

Answer 1

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.