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 $-3\left(2x2-x+1\right)=-6x2+3x-3$. 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 $2x3+x2+1$.

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, your turn. Try $2x^2-x+1$ times $2x^2 +3x -1$. Do you get this picture?

Do you see the answer $4x^4 + 4x^3 - 3x^2 +4x -1$?

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