Suppose \(g(x)\) is a polynomial of degree five for which \(g(1) = 2, g(2) = 3, g(3) = 4, g(4) = 5, g(5) = 6, g(6) = -113\) . Find g(0).

We can use a difference table on the values of the polynomials: https://brilliant.org/wiki/method-of-differences/

So the first differences are 1, 1, 1, 1, -119. The second differences are 0, 0, 0, -120.

If we continue, we find that g(0) = -17.

I agree with #1 that a difference table is the way to go, I don't see how to use the remainder theorem, but I get g(0) = 121.