Station 15
Resolution

Here’s the division problem from the last station: $\dfrac {x^{3}-3x+2} {x+2}$ .

And here is the picture for it in a $1 \leftarrow x$ machine.

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Let’s go!

Question 1

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We need to find all the copies of $x + 2$ (one dot next to two dots) anywhere in the picture of $x^{3}-3x+2$ , but we can't find any.

And we can’t unexplode dots to help us out as we don’t know the value of $x$ . (We don’t know how many dots to draw when we unexplode.)

The situation seems hopeless at present.

But I have a piece of advice for you, a general life lesson in fact. It’s this.

If there is something in life you want, make it happen!

(and deal with the consequences.)

Right now, is there anything in life we want?

Look at that single dot way at the left. Wouldn’t it be nice to have two dots in the box next to it to make a copy of $x+2$ ?

So let’s just put two dots into that empty box! That’s what I want, so let’s make it happen!

But there are consequences: that box is meant to be empty. And in order to keep it empty, we can put in two antidots as well!

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Brilliant!

Finish up computing $x^{3}-3x+2 \div x+2$ on the on the $1 \leftarrow x$ machine to make sure you can see how the app works.

(Hint: Can you make antidots have the pattern we want?)

Question 2

If you are looking for some practice problems, feel free to try these. Try them with pencil and paper, and then with the app perhaps.

Compute $\dfrac {x$

Question 3

Compute $\dfrac {4x$

Aside: Is there a way to conduct the dots and boxes approach with ease on paper? Rather than draw boxes and dots, can one work with tables of numbers that keep track of coefficients? (The word synthetic is often used for algorithms one creates that are a step or two removed from that actual process at hand.)

Once you’ve read this lesson, check out Levi’s video here for more explanation and practice!