# Intuition for Polynomial Interpolation Error

Math StackExchange, Wikipedia and Proof Wiki all offer roughly similar proofs of the error formula for polynomial interpolation of a $$n+1$$ differentiable function $$f(x)$$ with a $$n$$-degree polynomial $$p_n(x)$$:

$$f(x) - p_n(x) = \frac{f^{n+1}(t)}{(n+1)!} \prod_{i=0}^n (x - x_i)$$

The proofs proceed by constructing a new function that has n+2 roots and taking n+1 derivatives, and I can follow the proofs well enough. For example, the Math StackExchange link above defines the auxiliary function as $$F(x) = f(x) - p(x) - (f(\bar{x}) - p_n(\bar{x}))(\prod_{i=0}^n (x-x_i))^{-1}(\prod_{i=0}^n (x-x_i))$$.

I have two questions:

1. What motivates this auxiliary function? It appears that we're trying to give $$F(x)$$ an additional root, but why is this the correct starting step?
2. The error formula appears unintuitive to me. It links the interpolation error to a many-time-differentiated function multiplied by a polynomial over a factorial. How can I easily see why this error statement is true? What's a good way to explain why the interpolation error equals the right hand side?