Twelfth grader Abbey wants some help with the following: "Factor x⁶ +2x⁵ - 4x⁴ - 8x³ + x² - 4."

Well, Abbey, if you've read our unit on factoring higher degree polynomials, and especially our sections on grouping terms and aggressive grouping, you probably realize that a good way to attack this problem is to try grouping the terms. Hopefully, you tried something along those lines. Here's one method (I'll get you started on another method at the end of this post):

(x⁶ + 2x⁵) - (4x⁴ + 8x³) + (x² - 4).

That looks promising, since every group can be factored:

x⁵(x + 2) - 4x³(x + 2) + (x - 2)(x + 2).

Great! every group has an (x + 2) term in it, so we can factor that out:

(x + 2)(x⁵ - 4x³ + x - 2)

I'm going to attempt grouping on that second polynomial:

(x + 2)[(x⁵ - 4x³) + (x - 2)]

(x + 2)[x³(x² - 4) + (x - 2)]

(x + 2)[x³(x - 2)(x + 2) + (x - 2)]

And once again we have a common factor in our groups - this time it's (x - 2).

(x + 2)(x - 2)[x³(x + 2) + 1]

(x + 2)(x - 2)(x⁴ + 2x³ + 1)

Now we have something that can't easily be grouped, because it has 3 terms. So I decided to try something crazy: I added in x² and subtracted it back out again. Why? Take a look:

(x + 2)(x - 2)(x⁴ + 2x³ + x² + 1 - x²)

(x + 2)(x - 2)[(x⁴ + 2x³ + x²) - (x²- 1)]

(x + 2)(x - 2)[x²(x² +2x + 1) - (x + 1)(x - 1)]

(x + 2)(x - 2)[x²(x + 1)² - (x + 1)(x - 1)]

My groups each contain an (x + 1), so I can factor that out:

(x + 2)(x - 2)(x + 1)[x²(x + 1) - (x - 1)]

(x + 2)(x - 2)(x + 1)(x³ +x² - x + 1)

At this point, I tried a couple different groupings, and nothing seemed to work to factor (x³ +x² - x + 1). That doesn't mean it's not factorable, but I got a little suspicious, so I pulled out the Rational Root Theorem, which tells me that if that thing has any rational zeroes, they have to be either 1 or -1. Since neither of those are zeroes of the polynomial, that means it's unfactorable over the rationals. Therefore, the final answer is:

(x + 2)(x - 2)(x + 1)(x³ +x² - x + 1)

Just a couple notes on this, before I'm done:

1. I enjoy the process of trying to do grouping (and what I call "aggressive grouping") on polynomials in order to factor them. I think it's a good mental exercise. Nevertheless, I should point out that according to the Rational Root Theorem, if this polynomial has any rational zeroes, they are any of the following: -1, 1, -2, 2, -4, 4. Trying out those to see which work, and then using Synthetic Division to divide out the factors will guarantee you a solution, even if you can't figure out groupings to do.
2. There's another way to start off the problem (which is actually what I did the first time I solved the problem), which involves rearranging the terms before grouping: (x⁶  - 4x⁴) + (2x⁵- 8x³) + (x² - 4). Grouping this way has the advantage of pulling out (x + 2)(x - 2) all in one step, and going straight to a 4th degree polynomial. It's nice to see that the problem can be done in multiple ways, don't you think?

Second grader Phushisa from Thailand asks this question: "What is your favourite song?"

Thank you for asking that question, Phushisa! Normally I get asked questions about math, or about poetry, or science, and I don't usually get to talk about myself, so this is a fun question.

Before I begin, I should point out that there is a huge amount of music in this world, and I really love a lot of it, and listen to a lot of it, and it would be almost impossible to pick on single song. In fact, my favorite song varies from day to day and time to time, depending on my circumstances, my mood, etc.

So I'm not going to pick one single favorite. I hope that's okay.

Also, I'm going to stretch the definition of "song," because I'm a big fan of orchestral music - symphonies, concertos, and so forth. And no one calls those things "songs" - they call them "pieces" or "compositions" or (of course) "symphonies" and "concertos".

If I'm doing computer work, or preparing lessons for school, I don't really want my brain actively engaged in the music, so I will rarely listen to music with words. This is when I spend time listening to "classical" music. (I put the word "classical" in quotes because "classical" is an era of musical composition, but outside of the music education and performance community, people use the word "classical" to describe any orchestral or operatic music.)

Speaking of which, I'm not a fan of opera, just to get that settled.

So here are a couple of the pieces of orchestral music that I most love:

Tchaikovsky's Symphony #5
This is the one that I give as my standard answer when someone asks "What's your favorite symphony?" In general, I love anything by Peter Tchaikovsky, but this one tops the list for me. There are some absolutely beautiful, lyrical sections, and some sections that are full of drama and intensity. The last movement (section) keeps building and building, and just when you think it's almost done, it keeps building some more. If you'd like to hear it, there's a recording on YouTube of Leonard Bernstein conducting it. If you don't have time to listen to the whole thing, jump to the last five minutes, and watch how insanely excited Bernstein gets while conducting. It's worth a few minutes of your time.

Khatchaturian's Symphonic Poem #3
This thing is what I listen to when I feel like getting a little more modern. Aram Khatchaturian's music sometimes seems a bit discordant, and off-kilter, but for sheer power and drama, it's pretty amazing. This piece of music has an organ as one of the instruments, and the organist's fingers have to be flying across the console. I think it must be a very challenging piece for the entire orchestra, because I've listened to several recordings of it, and none of them sound at all like the others. My favorite is the Chandros recording, which can be found on YouTube as well. One commenter described the end of the piece as "insane, pure musical carnage," Good description.

If I'm not working, I'm more likely to want music with lyrics (singing), and there's so much music out there that rather than picking a favorite song, can I pick a favorite musician? I promise, it'll be interesting, because he's a musician most people I know haven't even haven't heard of. His name is...

Andrew Peterson
Peterson is what I would consider to be a real wordsmith. His use of language, metaphor, rhythm and rhyme ranges from elegant to almost comical, depending on what he's trying to say, and what he's trying to say is usually very thought provoking (even when he's funny). He's a master of slant rhymes and unexpected rhymes. I remember listening to one of his songs and hearing this line: "We've got Andy on the guitar 'cause I promised him Some Texas barbecue." And every time I heard that line, I felt like it was a pair of lines that rhymed, yet for the life of me I couldn't figure out why, because there are no rhyming words in there. Finally, I realized that the rhyme was the second syllable of "guitar" with the first syllable of "barbecue" ("tar" with "bar"). And I thought, "Who in the world comes up with something as crazy as that, and makes it work?"

And the answer is, obviously, "Andrew Peterson."

Have you ever noticed that sometimes you'll hear a song and really love it right off the bat, but gradually get tired of it? Peterson has the opposite effect on me; it usually takes me a few listens to really appreciate it musically and lyrically, but then it just keeps growing on me, and I don't tire of it easily.

Favorites? My wife says that one of her favorites from Peterson's most recent album is "Be Kind to Yourself," which Peterson wrote for his teenaged daughter (and then sang with her, which you can watch/listen to in the video linked).

From the same album, one of my favorites is "The Rain Keeps Falling," which is about finding both peace and hope for the future in the troubles of life. I'd also add this one, which has a similar theme: "The Sower's Song." If you listen to this one, pay attention to the percussion underlying the words. Usually the drummer gets to show his stuff during the pauses between lyrics (that's why they call them drum fills - because they fill the gaps between lines of words), but toward the end of this song there are some really interesting things happening with percussion that builds in intensity during the line of singing.

I hope you enjoy some of this music as much as I do!

"Can the context of a word change the way its syllables are stressed/unstressed? Myra"

Yes, Myra, the context can definitely change the way we stress the syllables. Here's a simple example:

"I would never desert you in the desert."

How did you stress the syllables in the word "desert"? Hopefully, the first time the word showed up, you pronounced it "de-SERT," with the stress (accent) on the second syllable. But what about the second appearance of the word? You didn't pronounce it the same way, did you? You pronounced it "DES-ert," with the accent on the first syllable.

The difference, of course, is that even though they are spelled the same, these are two quite different words. One of them is a verb, that means to abandon, and the second is a noun -- a dry and barren landscape.

There are occasions when a single word functions as either a noun or an adjective, depending on the context, and the difference in part of speech is reflected in the way it is pronounced. For example, consider the word "arithmetic." Is it a noun? Or is it an adjective? It actually could be either, depending on the context.

"I really enjoy studying arithmetic."

In this case, "arithmetic" is a noun, a branch of mathematics, and we pronounce it "a-RITH-me-TIC," with the primary accent on RITH, and a secondary accent on TIC.

"I really enjoyed studying an arithmetic sequence."

In this case, "arithmetic" is an adjective rather than a noun; it is describing a type of sequence. As an adjective, the word is pronounced "A-rith-ME-tic," with the stresses inverted from the noun pronunciation.

Can you think of other words that have different pronunciations depending on the context?

Incidentally, there's another way that context can affect how syllables are stressed. Sometimes, the innate rhythm of a phrase we're saying can lead to slight variations in accent. I emphasize here the word slight.

Here's one example. I mentioned the word "sequence" above; let's take that word and make it plural: "sequences." If you pronounce this word, you will likely pronounce it like this: "SEQ-uenc-es," with primary stress on the first syllable, and no secondary stress. I checked a phonetic spelling in an online dictionary, and that is exactly what it showed.

So now let's take the phrase "arithmetic sequence" and make it plural: "arithmetic sequences."

How do you pronounce this? The answer (I love combining math and poetry) is that you pronounce it iambically, which means you pronounce it with alternating unstressed and stressed syllables: "A-rith-Met-ic SEQ-uenc-ES."

Your sense of rhythm almost begs you to put a secondary stress on that final syllable. Now, as I said, it's very slight, and maybe you don't add that extra stress yourself, but I can tell you, as a math teacher, I hear that extra accent all the time. The same is true for the phrase "geometric sequences."

The same thing happens (for essentially the same reason) when we read poetry. Say the word "happy" and ask yourself where the stress is. Presumably you said that it comes on the first syllable: "HAPP-y."

But now read these very familiar lines of poetry: 

But I heard him exclaim, 'ere he drove out of sight,
"Happy Christmas to all, and to all a good night!"

Did you notice what happened? Your brain subconsciously (or maybe consciously) got so wrapped up in the rhythm of unstressed/unstressed/STRESSED over and over again that you really didn't give stress to either syllable of "happy," waiting instead to punch the first syllable of "Christmas."

In fact, that's precisely what the author intended you to do; it is a form of poetry called "anapestic tetrameter," in which every stressed syllable is preceded by two unstressed syllables. And those two unstressed syllables are the word "happy."

Related post: How to recognize stressed and unstressed syllables.

"Hi Professor Puzzler, in last week's post you said that if a number is irrational, it doesn't terminate or repeat. I wonder how you know that? Of all the irrational numbers, how do you know there isn't one that terminates or repeats?"

This question comes from Trevor in Georgia.

Hi Trevor, in answer to your question, I'll give you a short proof (with a bit of hand-waving over some details, just to keep the write-up from being too long!). What we'll do is, we'll start with numbers written in decimal notation, which we'll assume are irrational (which means, as you remember from the post about the square root of two, that it can't be written as a ratio of integers) and then we'll arrive at a contradiction, which will prove that they aren't irrational. So let's get started!

Terminating Decimal

Let n be an irrational number with a terminating decimal. This means that after a certain number of decimal places (let's call that k), the decimal ends.

Since the decimal terminates after k decimal places, the number 10^kn is an integer (every time you multiply by 10, you shift the decimal one place to the right, so multiplying n by 10^k results in all the digits being to the left of the decimal).

Now we can write n = (10^kn)/(10^k), which is ratio of two integers, and that contradicts our original assumption that n was irrational. Thus, if a number in decimal form terminates, it is not irrational.

Repeating Decimal

Let n be an irrational number with a repeating decimal. This means that after a certain number of decimal places (let's call that k), the decimal begins repeating every h digits, where h is some integer.

For example, if the number is 3.125727272..., k = 3, h = 2.

To simplify our math, let's multiply our number by 10^k, which allows us to get the non-repeating part of the decimal over to the left of the decimal. We understand that when we're done with our manipulations, we have to divide by 10^k to get our number back, but if we've proven that our result is rational, then dividing by 10^k still gives us a rational number.

Similarly, everything to the left of the decimal can then be subtracted, with the understanding that when we're done we must add it back in. But the same argument holds; if we've proved the result is rational, adding an integer to that rational number will result in another rational number.

In other words, we can, without loss of generality, simplify our starting point to a decimal which looks like this:

n = 0.a₀a_1...a_ha₀a_1...a_ha₀a_1...a_h...

Let m be the integer formed by the following digits: a₀a_1...a_h

Then n = m/(10^h) + m/(10^2h) + m/(10^3h) + ...

This is an infinite series with common ratio 1/(10^h), and its sum is:

S = [m/(10^h)]/[1 - 1/(10^h)]

S = m/(10^h - 1)

Since m is an integer, and 10^h - 1 is an integer, we've shown that our number can be expressed as a ratio of integers, which contradicts our assumption that it is irrational. From this we've shown that if a decimal repeats or terminates, it is not irrational.

Thanks for asking, Trevor!

"In your post about triangular mountains, you mentioned ice wedging - can you explain what that is?"

Sure! In fact, I'll do better than just explain; I'll also give a picture. First the explanation. Ice wedging is a form of erosion. And erosion, in case you don't know, is when something breaks down due to wind, water, or other natural forces. Erosion can easily be seen on a mountain trail, where water rushes down the side of the mountain (usually in the spring time, but after any heavy rain) and washes away the dirt. This is why, if you're hiking a mountain, you'll often find trails covered with tree roots. The roots weren't always exposed; the rain washed away the dirt until they were sticking out.

This is also why, on many steep mountain trails, the people who maintain the trails will put logs across the trail at intervals, diagonally placed in such a way that water rushing down the trail will be diverted off into a ditch by the side of the trail. These kinds of devices are called "waterbars," because they are designed to bar the water from flowing down the trail and eroding it.

Okay, so that's erosion, and "ice wedging" is a form of erosion. Ice wedging happens when a rock has a crack in it. When it rains, the crack fills up with water. Now, that's not really a big deal, unless that water freezes inside the crack. Since water expands when it freezes, there's not enough space for all that ice in the crack, so the two sides of the crack get pushed apart by the ice, which causes the boulder to shift, and maybe crack some more. If this process happens repeatedly, the crack gradually becomes bigger and bigger, and eventually the boulder may split entirely.

This is very common at high altitudes, where the temperatures fluctuate a lot from day to night; during the spring time there will be lots of rain, and lots of running water from melting snow, but at night the temperatures drop below freezing.

Here's an example of a boulder near the summit of Cranberry Peak in Stratton, Maine, that was probably split due to ice wedging.