Let a > 0, and let P(x) be a polynomial with integer coefficients such that, P(1) = P(3) = P(5) = P(7) = a and P(2) = P(4) = P(6) = P(8) = -a. What is the smallest possible value of a?
I have no idea how to do this. So if a mathematician wants to explain it that would be really good.
Of course, if the coefficients did not have to be integers then a could be as close to 0 as you please.
I did construct a graph of degree 7 but it is not actually helpful
Here is a link:
Using Mathcad to do the heavy lifting I get the following:
I suspect there is a more elegant way, but I can;t think of one!