Blog: Difference of Squares, Cubes and Quarts

I wanted to do this before the Algebra Regents exams when it might have been more helpful to some students, but the fact is, they don't actually follow me. They just find the posts with the questions and answers on them through search engines.

The Difference of Squares is a common topic for Algebra 1, but what about other powers, such as cubes and quarts? (Quarts are is "quartic", fourth power.)

Squares and Quarts

In a binomial that looks like x² - n², where n is any number, which will be squared, we can factor the two terms into conjugates:

(x + n)(x - n)

All nice and neat. When you multiply the two conjugates (which have the same terms, but one has a plus and one has a minus), the two x terms -- (nx) and (-nx) -- are additive inverses and will sum to zero.

Every now and then, state exams like to throw a curve ball at you, and use a higher power, usually 4.

What if the problem said:

Factor completely x⁴ - 16
Since 4 is an even power, x⁴ is a perfect square. And, of course, 16 is a perfect square. So the regular rule applies:
x⁴ - 16 = (x² + 4)(x² - 4)
Ah, but have we factored "completely" as the question asked? We can't do anything with the first factor, (x² + 4), which has no real roots, but what about the other factor? It's another Difference of Squares, so we can apply the rule again:
x⁴ - 16 = (x² + 4)(x² - 4) = (x² + 4)(x + 2)(x - 2)

So, if they really wanted to be mean, er, I mean "challenging", they could go another step further and ask:

Factor completely x⁸ - 256

Sum of Cubes

I know that I started by saying "difference of ...", but that was to keep everything flowing.

Unlike squares (and other even powers), you can factor the sum of two cubes. Also unlike squares, a perfect cube can be a negative number, so it also could be written as a difference of cubes. So I'm not wrong -- I just renamed it for no reason.

There's a simple format for the Sum of Two Cubes:

(x³ + n³) = (x + n)(x² - nx + n²)

First thing you should see is that (x + n) means that there is a root at x = -n.

The second thing you should see is that that is the only real root. If you check the discriminant, b² - 4ac, you get the following:
(n)² - 4(1)(n²).

As long as n has a nonzero value, the discriminant will always be less than zero. This example had a leading coefficient of 1, but it is still true if there is a different number in front the x³ term.

Final Example

If we wanted to put this together, we could try to factor x⁶ - 64.

Or really put it all together and got with an x¹² term. But who wants to do that?

Day 24 of 30: Rubik Had a Cube Long Before the Borg Did

This is Day 24 of the 30-day blogging challenge. The finish line is in sight.

Rubik had a cube long before the Borg did. Oh, sure, the Borg Collective might be an ancient race, or perhaps just an extremely old one, but we didn't sight any of their cube until sometime in the 90s, and that was glancing three or four centuries into the future even with that. Rubik had a cube while I was in high school. According to what I've read today, Rubik's Cube is 40 years old. This is a little surprising, because that puts its invention in the early 70s (1974, obviously ... I know, I'm a math teacher!), but the cube as a phenomenon didn't occur until the early 80s, at least seven years later. I remember. I had them in high school. My brother even set me up in business once at a block party, solving them for a quarter a pop. Actually, I only made fifty cents because two of the cubes were unsolvable. And I had to prove it to them.

Unsolvable cube Number One was completely finished except for two corners which were rotated out of place. They couldn't be rotated back into place. Rotating them involved a minimum of three corners. That's just the math. To move one thing, something else had to be moved. There was no way that those could be moved. Why? "This cube has been taken apart and put back together." I didn't say that they had done it or that it had been done on purpose. In fact, knockoff cubes fell apart fairly easily because they didn't spin well. If you didn't completely twist, say, the front face and then tried to turn the top face, a corner or two might fly off. Should this happen, someone would scoop up the loose piece and snap it back in. The problem is that you had a two of of three chance of snapping it in incorrectly. At this point, it would almost have been better just to have taken the entire cube apart and resemble it in the original state of six solid faces. But that was a daunting task. Snapping in one or two loose pieces didn't pose a big problem. Reassembling the entire cube could be problematic. I don't know anyone who ever tried it.

It took a while to explain, but they finally accepted my explanation of Unsolvability. In retrospect, I should've charged her the quarter anyway. I did solve it as far as it could be solved.

Unsolvable cube Number Two: another knockoff. It didn't have the "Rubik's" logo and it had the wrong color scheme. On a real cube, white was opposite blue, yellow opposite green, red opposite orange. (I wrote that from memory without checking ... hope I'm right!) Knockoff could have colors anywhere. This one did. To make matters worse, a couple of the center pieces had fallen off. The center of a face didn't really go anywhere. Yes, you could move them around, but they were essentially the plus and minus ends of the x-, y-, and z-axes. They determined what the other eight square around them should be.

Now this didn't pose a problem for me because I was a high school teenager and, therefore, a Super Genius. She had one side completely done, all yellow. However, they weren't in the correct places. You see, it's not enough that all the yellows are on one face, but the corners have to be in the correct positions as well as the edges so that they line up with the center pieces on the adjacent sides. I looked to see which colors yellow was touching so I could figure out which color was on the opposite side from yellow, by process of elimination. The opposite side wouldn't touch yellow. Except that they all did. He had peeled off the stickers and put the yellows on one side. Going around the top ring, yellow touched all five other colors, which is impossible. Thankfully, that didn't take much time or effort to figure out, so I didn't feel like I was cheated out of a quarter (not that he was going to pay me in advance -- he didn't believe I could do it even after I'd done two.)

My method wasn't one of the quick ones, and I never solved a cube in under two minutes. Forget about the 25-second methods. They weren't going to happen. I'd have to relearn. At that point, it was all about color recognition and fast reflexes. That wasn't going to happen. I was happy enough to know how to solve one. (And thanks to Bro. Steve for handing out the instructions in our Number Theory class.

One final note: I like to play a game called Fluxx produced by Looney Labs, founded by Andrew and Kristin Looney (no, really). I discovered online from one of her friends, that Kristin (not yet Looney) was a Regional Rubik's Cube champion on TV's That's Incredible years ago. Here's the link. Enjoy.

Cube Square

(Click on the cartoon to see the full image.)

(C)Copyright 2014, C. Burke.

When they tell you that you can't have giant ants, they mean that you can't mutate a regular ant into a giant one. But could they evolve that way on their own?

Actually, that was the point of a SF story I read years ago, I think, in one of the Dangerous Visions anthologies.

This isn't the first time giant ants have appeared in (x, why?). For more giant ants, check out Ants vs. Spiders.

And then there are these:

R's Cube

(Click on the cartoon to see the full image.)

(C)Copyright 2010, C. Burke. All rights reserved.


Not to imply that grading standards, while somewhat adjustable, are totally interchangable just by rotating your position 90 degrees ...