Can we multiply polynomials? You bet!
Here’s the polynomial .
If we want to multiply this polynomial by 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 is .
Suppose we wish to multiply by instead. This means we want the anti-version of tripling all the quantities we see. So each dot in the picture of is to be replaced with three antidots and each antidot with three dots.
We have . We could also say that is the anti-version of .
Now suppose we wish to multiply by . Since looks like this
we need to replace each dot in the picture of 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 equals .
Now let’s multiply with , 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 .
Okay, your turn. Try times . Do you get this picture?
Do you see the answer ?
See how Goldfish & Robin multiply polynomials in this video where kids explain math to kids.