Lagrange polynomials sum to one

Question

I've been stuck on this problem for a few weeks now. Any help?

Prove: $\sum_{i=1}^{n}\prod_{j=0,j\neq i}^{n}\frac{x-x_j}{x_i-x_j}=1$

The sum of lagrange polynomials should be one, otherwise affine combinations of with these make no sense.

EDIT: Can anybody prove this by actually working out the sum and product? The other proofs make no sense to me. Imagine explaining this to someone who has never heard of lagrange.

Answer 1

Let $P$ the polynomial with degree $n$ defined by $$P(x)=\left(\sum_{i=0}^{n}\prod_{j=0,j\neq i}^{n}\frac{x-x_j}{x_i-x_j}\right)-1$$ then we see easily that $$P(x_i)=0,\;\quad\forall i=0,\ldots,n$$ so $P$ has $n+1$ distinct roots hence it's the zero polynomial due to the D'Alembert's theorem. Conclude.

Answer 2

HINT: Throw in the $f(x_i)$ and what happens if they are all 1?