Binomials Here, Binomials There, Binomial Theorems Everywhere, Part 1

I was recently asked to explain just how the Binomial Cube demonstrates the binomial theorem. So let's start not with the cube, but with the binomial theorem. If this part means nothing to you, just skip ahead to the part where I start describing the material, and later this part will make more sense.

What is the Binomial Theorem?

In middle or high school algebra, you probably learned (or "learned") a pattern that went like this:

$$(a + b)^2 = a^2 + 2ab + b^2$$

This is the very simplest case of the binomial theorem. The binomial part has to do with the fact that there are two ( "bi- " ) terms ( "nomen"), namely a and b, which are added together. The theorem gives us a pattern for finding the second power (square) of the sum $a + b$, even if we don't actually know what a and b are.

So what does this have to do with the Binomial Cube?

Technically, nothing. It has to do with the lid of the binomial cube, but we’ll get to that.

If you had a very traditional mathematical upbringing, the explanation above might be all you ever learned about the binomial theorem, and it probably drove you nuts. I’m guessing that you frequently made this mistake in your algebra:

$$(a + b)^2 = a^2 + b^2$$

Looks reasonable, doesn’t it? But it’s not true. Here’s a straight up arithmetic example to see why. Let’s make $a=3$ and $b=4$. Then $(a + b)^2 = (3 + 4)^2 = 7^2 = 49$. On the other hand, $a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25$. That’s a long way from 49. To see why, let’s draw a picture of what’s going on. On the left is $7^2$ (The colors here match the Montessori bead bar colors, but there’s no real significance to the colors). On the right is $4^2$ and $3^2$, which I’ve put inside a 7-square so that we can see what the problem is:

There is extra space left in the square on the right. We’re going to have to fill those in somehow. So let’s see what we can do:

No we can see that we can fill these corners in with three 4s (the top left) and four 3s, or $4 x 3$ and $3 x 4$. Since these are the same thing, we can just say that this is $3 x 4$ twice. So now let’s see what we’ve worked out:

$$(3 + 4)^2 = 3^2 + 2(3 x 4) + 4^2 = 7^2$$

This is just the binomial theorem applied to these particular numbers.

Let’s get back to the binomial cube. What I just demonstrated is a variation on the earlier lessons we present to children to introduce them to the binomial theorem. (I did it a bit backwards, but the idea will suffice for now).

The rectangular pattern you can see above may remind you a bit of the lid of the binomial cube box. That’s not an accident. Here’s a picture of the box:

The Pattern on the Lid of the Binomial Cube

Looks a lot like the first picture we drew. So now, let’s label the sides. Instead of making them stand for specific numbers though, we’ll just call the parts $a$ and $b$, like this:

So now, each side has a length of $a$ and $b$ put together, or $a + b$, and we can look at the boxes as geometrically representing multiplication, just like they did in our arithmetic version above:

(Note, when we switch to symbols, we stop using a multiplication sign, so $a x b$ is written as $ab$. This is something I normally explain to the children when I first introduce some algebraic ideas.) Now, we can put this together in symbols:

$$(a + b)(a + b) = (a + b)^2 = a^2 + 2ab + b^2$$

Huzzah! We have the binomial theorem.

Tune in next time, when we talk about how to use the binomial cube to go further.