# Multiplicity

We continued our exploration of polynomials today. We ended last time with them struggling to graph something like $f(x) = 3(x-1)(x+2)^2$. They knew how to find the intercepts and the end behavior, but we hadn’t discussed multiplicity. There were some wide eyes. Some “hey, wait!”s. It was good. They started seeing that they didn’t know the full story.

Today, we started with a similar problem. I like what I ended up doing more than what I had in my notes, so this is my new notes.

We started by graphing $f(x) = 3x^3 + 12x^2 + 12x$, which gave us a chance to talk about how we should always factor out the GCF. Then we plotted the x-intercepts and the end behavior, and I asked, “So, how do we connect the dots?” I handed the marker to the first kid to start talking, and we discussed it: should it cross through at x = -2 or at x = 0? Should it look like a line or a parabola near these points? Why?

I told them that near the x-intercept, the function looks like the factor that it came from – and that’s the part that I liked and that I didn’t. I liked this more than just “Oh, look, there’s a pattern!” But I probably should have talked about why, how the numbers so close to zero overpower everything else, so this factor overpowers everything else. Next time.

But since we aren’t totally sure what $(x-3)^{17}$ looks like, we needed to explore. That looked different in different classes. In two classes, we graphed on Desmos all the powers of (x-3) until someone noticed a pattern: odd powers cross the axis, even powers bounce off. In another class, I told them to graph their favorite power of (x-3) and find everyone that looked like them. This may have actually backfired, because $(x-3)^{501}$ looks pretty flat, so it may have made them think you can have a lot of x-intercepts. I’ll have to watch out for that.

After some exploring, we eventually landed on a pattern, then started graphing these things. That started building the need for the word “multiplicity,” because I said “exponent on the corresponding factor” a lot, which got long and unwieldy. I liked that I didn’t give them the word right away – we saw why the concept was important and how it was painful to talk about without the word.

I’ve been doing a lot of “graph this on Desmos, and find everyone with a graph like yours; what do you have in common?” lately. I like it, because it makes them talk instead of me telling them; it also means that more people are engaged than if we went through it as a group with just one or two people calling out the answers. Desmos is also far, far superior to the graphing calculator. It’s a little slow to load, but I think it’s worth it for them to be talking instead of me.