Exploding Dots

1.8 A Problem?

Lesson materials located below the video overview.

Here’s the problem from the end of the previous section:

$$\dfrac{x^{3}-3x+2}{x+2}$$.

Is this problem a problem?

Up to this point we’ve been presenting examples with all coefficients positive numbers. This is our first example with negative coefficients.

Here’s what the set-up for this  division problem looks like in our $$1 \leftarrow x$$ machine:

Our task is to find groups of  within the top diagram, and right away matters seem problematic. One might think to “unexplode” dots to introduce new dots (or anti-dots) into the diagram but there is a problem with this: We do not know the value of $$x$$ and therefore do not know the number of dots to draw for each “unexplosion.”

There is a way to cope with this difficulty. It requires a truly wondrous epiphany.

IF YOU HAVE NOT HAD THIS EPIPHANY FOR YOURSELF, PAUSE BEFORE READING ON. See if a wondrous epiphany comes your way.

Ask: What would I like to see in this picture that would, at least, get me started? Can I make what I want to appear actually appear?

Here’s the epiphany:

If there is something you want it life, make it happen! (And deal with the consequences of your actions.)

I would love to see two solid dots in the second box from the left, to match the single dot in the leftmost box. But I can’t just add two dots to the problem without changing the problem … but I can if I counteract that action with the addition of two anti-dots as well! This keeps the value of the box zero.

And if I stare at this picture I also see an anti-version of  at the $$x$$-level. Whoa!

Carrying on with my “make it happen” approach, let me add a solid dot (and anti dot) to the second-to-last box.

We see:

$$\dfrac{x^{3}-3x+2}{x+2} = x^{2} – 2x + 1$$.

(We can check this by multiplying $$x^{2} – 2x + 1$$ by $$x+2$$.)

AMAZING!

 Question 42: Use dots-and-boxes to compute the following: a) $$\dfrac{x^{3}-3x^{2}+3x-1}{x-1}$$ b) $$\dfrac{4x^{3}-14x^{2}+14x-3}{2x-3}$$ c) $$\dfrac{4x^{5}-2x^{4}+7x^{3}-4x^{2}+6x-1}{x^2-x+1}$$ d) $$\dfrac{x^{10} -1}{x^{2} – 1}$$   Challenge: 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.)

Recall that all solutions appear in the COMPANION GUIDE to this EXPLODING DOTS course.

 Question 43: Use an $$1 \leftarrow x$$ machine to compute each of the following:   a) $$\dfrac{x^{2} – 1}{x – 1}$$   b) $$\dfrac{x^{4} – 1}{x – 1}$$     c) $$\dfrac{x^{6} – 1}{x – 1}$$     d) Will $$\dfrac{x^{even} – 1}{x – 1}$$ always be a multiple of $$x-1$$?     e) Compute $$\dfrac{x^{6} – 1}{x + 1}$$.     f) Will $$x^{even} – 1$$  is also always be a multiple of $$x + 1$$?     g) Explain why $$2^{100} – 1$$ must be a multiple of $$3$$ and be a multiple of $$5$$. (HINT: Let $$x = 2^{2}$$. Then $$2^{100} – 1 = x^{50} – 1$$.)  Show that it is also a multiple of $$33$$ and of $$1023$$.     h) Is $$x^{7} + 1$$ divisible by $$x-1$$? Is it divisible by $$x + 1$$?     i) Is $$x^{odd} + 1$$ a multiple of $$x-1$$? Of $$x+1$$?     j) Explain why $$2^{100} + 1$$ is a multiple of $$17$$. Show that $$3^{100} + 1$$ is a multiple of $$41$$.

 Question 44: REMAINDERS a)    Using an $$1 \leftarrow x$$ machine show that $$\dfrac{4x^{4} – 7x^{3} + 9x^{2} -3x -1}{x^{2} – x + 1}$$ equals $$4x^{2} – 3x + 3$$  with a remainder of $$2x-3$$ yet to be divided by $$x^{2} – x + 1$$. (This means: $$\dfrac{4x^{4} – 7x^{3} + 9x^{2} -3x -1}{x^{2} – x + 1} = 4x^{2} – 3x + 3 + \frac{2x-3}{x^{2}-x+1}$$.)     b)    Compute $$\dfrac{x^{4}}{x^{2} – 3}$$.     c)     Compute $$\dfrac{5x^{5}-2x^{4}+x^{3}-x^{2}+7}{x^{3}-4x+1}$$.     HINT: Drawing dots and anti-dots in cells is tiresome. Instead of drawing 84 dots, as you will need to do at one point for problem c), it is easier just to write “84.”

Comment: As a high-school teacher I do teach Exploding Dots for polynomial division in my algebra classes – the benefits of the conceptual understanding are too strong to ignore! However, this dots-and-boxes method is not friendly to pencil-and-paper. For this reason, once conceptual understanding firmly in place, I introduce a second method for computing polynomial divisions, one still rife with conceptual understanding, but friendlier on the pencil. It is the Reverse Galley Method explained in lesson 2.3 of the ASTOUNDING POWER OF AREA course on this site.

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