I've been having difficulty understanding this problem for awhile and figured out why. To clear things up, I just need to understand the logical difference ( if there's no difference, then the problem is obviously missing information) between these sentences:
(Find a polynomial function $f_i$ of degree n-1 which is 1 at $x_i$ and 0 at $x_j$ for j $\ne$ i.),
(Find a polynomial function $f_i$ of degree n-1 which is 1 at $x_i$ and 0 at $x_j$, for j $\ne$ i.).
The placement of the comma after $x_j$, how does that differentiate the last sentence from the first?
It's not clear to me how the significance (or lack of significance) of the comma is supposed to relate to the meaning (or lack of meaning) of the sentences. But here is how I would interpret the statement about the polynomial function $f_i$:
There are $n$ numbers of interest to us, $x_1, x_2, \ldots, x_n$. No two of these numbers are the same.
The polynomial function $f_i$ is of degree $n - 1$.
$f_i(x_i) = 1$.
For each $j$ in $\{1,2,\ldots,n\}$, if $j \neq i$ then $f_i(x_j) = 0$.