In the text Gautschi. Numerical analysis: an introduction, Birkhäuser, Boston, 1997; 2nd edition, 2012, Gautshi gives the following motivation for error in polynomial interpolation:
"It is not difficult to guess how the formula for the error should look: since the error is zero at each $x_i$, $i=0,1,...,n$, we ought to see a factor of the form $(x-x_0)(x-x_1)\cdots (x-x_n)$."
This part makes sense. He goes on to say
"On the other hand, by the projection property 3"[he is referring to the fact that if $P_n$ is the linear operator taking a continuous function to its $n$th degree interpolant, then $P_nf=f$ if $f$ is itself an $n$th degree polynomial] " the error is also zero (even identically so) if $f\in \mathbb{P}_n$, which suggests another factor-the $(n+1)$st derivative of $f$."
I begin to feel shaky here. I think what he is saying is that we know that the error is exactly $f^{(n+1)}$ evaluated at some point since the polynomial vanishes. I am uncertain though. He then says
"But evaluated where? Certainly not at $x$ since $f$ would then have to satisfy a differential equation. So let us say that $f^{(n+1)}$ is evaluated at some point $\xi=\xi(x)$, which is unknown but must be expected to depend on $x$. Now if we test the formula conjectured on simplest nontrivial polynomial $f(x)=x^{n+1}$, we discover a factor of $1/(n+1)!$ is missing. So our final guess is $f(x)-p_n(f;x)=\dfrac{f^{(n+1)}(\xi(x))}{(n+1)!}\prod_{i=0}^n(x-x_i), x\in [a,b]$. "
At this point my questions are: What differential equation would $f$ have to satisfy? Why is it obvious that $\xi$ would depend on $x$? And How is he testing his guess with $f(x)=x^{n+1}$?