Exploding Dots

6.7 (Optional) Multiplying, Adding, and Subtracting Polynomials

See how Goldfish & Robin multiply polynomials in this video where kids explain math to kids.

Can we multiply polynomials? You bet!

Here’s the polynomial $$2x^2-x+1$$.

If we want to multiply this polynomial by $$3$$ we just have to replace each dot and each antidot with three copies of it. (We want to triple all the quantities we see.)

We literally see that $$3\left(2x^2-x+1\right)$$ is $$6x^2-3x+3$$.

Suppose we wish to multiply $$2x^2-x+1$$ by $$-3$$ instead. This means we want the anti-version of tripling all the quantities we see. So each dot in the picture of $$2x^2-x+1$$ is to be replaced with three antidots and each antidot with three dots.

We have $$-3\left(2x^2-x+1\right)=-6x^2+3x-3$$. We could also say that $$-3\left(2x^2-x+1\right)$$ is the anti-version of $$3\left(2x^2-x+1\right)$$.

Now suppose we wish to multiply $$2x^2-x+1$$ by $$x+1$$. Since $$x+1$$ looks like this

we need to replace each dot in the picture of $$2x^2-x+1$$ with one-dot-and-one-dot, and each antidot with the anti-version of this, which is one-antidot-and-one-antidot. (This is now getting fun!)

After some annihilations we see that $$\left(x+1\right) \times \left(2x^2-x+1\right)$$ equals $$2x^{3}+x^2+1$$.

Now let’s multiply $$2x^2-x+1$$ with $$x-2$$, which looks like this.

Each dot is to be replaced by one-dot-and-two-antidots, and each antidot with the opposite of this.

We see $$\left(x-2\right)\left(2x^2-x+1\right)=2x^{3}-5x^2+3x-2$$.

Okay, you’re turn. Try $$2x^2-x+1$$ times $$2x^2+3x-1$$. Do you get this picture? (I’ve not colored it this time!) Do you see the answer $$4x^{4}+4x^{3}-3x^2+4x-1$$?

The process here is no different from the long multiplication of lesson 3.6, but one might have to contend with antidots now.

Adding and subtracting in base $$x$$ is just like adding and subtracting in base $$10$$. And it is easier in fact! Since we don’t know the value of $$x$$ we will never explode dots. That is, we never need to perform “carries” as one does in base $$10$$ arithmetic!

We can draw dots and boxes pictures of these in an $$1 \leftarrow x$$ machine if we like.

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