can i assume this case for the problem: the first point is below the x-axis, the second is above, the third is below and so on
then 2 points would mean a degree of 1: it's just a line crossing the x-axis once
3 points would mean a degree of 2: function has to cross x-axis twice
4 points would be a degree of 3 (crossing the x-axis 3 times)
continuing with this logic, a function of 40 points, would have the smallest possible degree 39?
Your conclusion is correct, but the logic is not.
You must not make any assumptions about the list of 40 points because that list could contain
any 40 points in the xy-plane. (The only requirement is that each point on the list has a different x-coordinate).
The graph of a polynomial passing through those 40 points does not need to cross the x-axis. It's also possible that all of the points lie within a single Quadrant.
We could reword the first sentence in the exercise to read:
"Suppose you are given a list containing
any possible set of 40 points in the plane, each with a distinct x coordinate."
I think the point of this exercise is for students to visualize the general pattern. If you have n points (where no two points lie on a vertical line), then you're guaranteed that a polynomial of degree n-1 exists, passing through each point.
For some of these possible lists of n points, there
might exist one or more polynomials of degree less than n-1, but it's
not guaranteed. This exercise requires that you can
always find a polynomial, regardless of the given points.
Wikipedia states that this is proven by the Unisolvence Theorem, but I've never seen or heard of that theorem in any algebra classroom. Instead, a pattern is usually invoked, and the conclusion is simply given as a fact to memorize.
What is your class currently studying? Are you solving systems of polynomial equations? :cool: