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.