I was helping a high-school student with algebra homework about polynomial long division, which is straightforward for single-variable polynomials (and actually even easier than the numerical case, because there's no carrying involved in figuring out how much your divisor goes into the intermediate remainders); however, this student then put up something devised by the teacher:

(15x^4-y^2)/(x^2+y)

The problem is that it's not obvious to see where you should stop, and the final remainder isn't unique; rather it depends on how you order the variables, and this result is usually only covered in grad school (notably, it did not come out of the textbook but rather was devised by the teacher).


After trying my hand at it and not getting anything simpler than

15^2-15y+14y^2/(x^2+y)

I was disturbed because I wasn't getting a remainder of smaller degree than the divisor, so I posed a question here (after having found a Wikipedia article and something on Karl's Calculus Tutor) and saw why some generalizations aren't such a good idea to introduce to the early grades (even though I am generally all for it, like I think the single-variable case ought to be taught as early as the end of elementary school, shortly after getting over "using letters to stand in for numbers" at the beginning of pre-algebra, because it really is that straightforward): http://math.stackexchange.com/questions/32070/what-is-the-algorithm-for-long-division-of-polynomials-with-multiple-variables

[Edited on April 10, 2011 at 4:57 PM. Reason : gender-neutral language ITT

4/10/2011 4:56:51 PM

is it possible the teacher was well aware of all of this and was going to use it as some sort of teachable moment? Or to teach a concept other than simply solving the problem?

4/10/2011 5:53:52 PM

Maybe you just suck at math. Aren't you a chick wannabe or something? Method tranny ITT

4/10/2011 6:48:24 PM

^^In a graduate class on commutative algebra I could see it as such, but not in high school, where the relevant terms like "ideal" and "polynomial ring" and "Gröbner basis" have not yet been dealt with.

^I got into grad school in math, so I don't suck at it; also I'm not transgender, so fuck off

4/10/2011 7:56:10 PM

OMG YEAH THAT TEACHER IS TERRIBLE!

4/10/2011 8:08:33 PM

He's probably very good, going in the direction I would if I were a high-school teacher; however, this generalization isn't as nice and neat as many that could be fruitfully mentioned early, and I think he went over his head...

4/10/2011 8:18:43 PM

I had a teacher in HS pull something like that in precalc, teaching us how to derive with without normal calculus, by using the definition of a derivative. 4 full chalk boards later, and calling in the calc teacher, we gave up on the thing. Calc teacher couldnt even solve via normal calculus.

4/10/2011 9:00:43 PM

That shit is actually pretty cool, as long as the problem is chosen carefully; I somehow doubt that you would easily find the derivative of e^x by typical symbolic manipulation, for example.

4/10/2011 9:01:58 PM

I've seen worse. Lots worse for handing out course inappropriate homework. I've seen 3rd graders with homework that should be in pre-algebra.

4/10/2011 10:32:56 PM

Do you mean word problems more easily solved with algebra, like "In 5 years, Jimmy will be twice as old as Bobby, and now he's 4 times as old; how old is Bobby now?"
...or maybe "If Jimmy can mow 2 lawns in 3 hours and Bobby can mow 3 lawns in 4 hours, how long will it take Jimmy and Bobby to mow the same lawn together?"

Or are you talking about the raw introduction to symbolic manipulation characteristic of pre-algebra?

[Edited on April 10, 2011 at 10:54 PM. Reason : made the age problem have a reasonable answer

4/10/2011 10:47:10 PM

I remember a slew of word problems like that in the years before algebra. For instance, one boiled down to x+2=y; x+y=150 or something. The teachers just let everyone guess and check and I remember thinking that it was a stupid way of approaching the problem.

4/11/2011 11:07:14 AM