Differences of squares and cubes

My older son is working his way through Art of Problem Solving’s Algebra book and has come to a section about factoring sums and differences of cubes.  This topic is new to him and I thought we’d work through a few introductory examples with numbers before diving in today.

There are a couple of surprises, I think.  First, although x^2 - y^2 is easy to factor, x^2 + y^2 is not.  Second, when you move up to cubes it turns out that x^3 - y^3 is reasonably easy to factor after you play around for a bit, and x^3 + y^3 is, too.

I wanted to show him a bit about what was going on before he dove in this morning.

Here are our short talks:

(1) We started by talking about x^2 - 1


(2) From there we moved on to x^2 + 1 and found a lot of primes


(3) Next up was x^3 - 1 which we were able to factor with a little work.


(4) Finally, we looked at x^3 + 1 which actually did factor in a very similar way to x^3 - 1!


So, hopefully a useful introduction. I’d like to do a few more projects over the course of the week to help give some different perspectives on factoring differences of squares and cubes equations.