I have been given this question, and I am unsure what the intuition behind answering it would be, nor where to start:
Let $x_1 < x_2 < \dots < x_n$. Show that, for some given function $f$, the polynomial $$P(x) = \sum_{k=1}^n f(x_k) L_k(x), \quad\text{where}\quad L_k(x) = \prod_{i=1\\i\neq k}^n \dfrac{x-x_i}{x_k-x_i}$$, satisfies $$P(x) = f(x) \quad \text{for} \quad x=x_1, x_2, \ldots, x_n.$$
How would one approach this question, and what does it intuitively show by answering it?
You will find that $L_k(x_j)=\delta_{jk}$ with the Kronecker delta. The zeros originate with zeros of the linear factors in the defining product of $L_k(x)$, while the ones result from cancellation in the quotient factors of the product.
So in the end you compute $P(x_j)=\sum_k\delta_{kj}f(x_j)=f(x_j)$.