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 is easy to factor, is not. Second, when you move up to cubes it turns out that is reasonably easy to factor after you play around for a bit, and 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

(2) From there we moved on to and found a lot of primes

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

(4) Finally, we looked at which actually did factor in a very similar way to !

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.