The first steps into polynomials

We are still a bit on the jet-lagged side from Christmas vacation. I’ve been trying to take it easy because we seem to have about 30 min less than we usually have. Since we are looking mainly at AMC 10 problems right now, there’s always something interesting to look at even if we don’t have a lot of time.

Today this problem about quadratic equations gave him some trouble. It was interesting to me to see how much of a challenge it was for him to come up with an equation of the form $x^2 – px + 2 = 0$ where he knew the roots. It is an interesting step from “what is the product of the roots in this equation?” to “Give me a specific example of an equation where you know the roots and the product of those roots is 2.”

Here’s the problem we were discussing:

Problem #14 from the 2006 AMC 10 B

Let a and b be the roots of the equation x^2-mx+2=0. Suppose that a+(1/b) and b+(1/a) are the roots of the equation x^2-px+q=0. What is q?

Here’s the first part of this morning’s talk:


and the second part: