If the question is “Are there useful applications for polynomial division?” the answer is a definite yes! The hot air balloon problem isn’t one them, but there are other good applications, like both of you mention. I love the running time application (Jason). I also love the Lagrange interpolation application (Bowen).

I thought I’d add to what Bowen said about Lagrange interpolation, and show that it really is intimately related to polynomial division. Some people maybe haven’t thought about this before: Lagrange interpolation is equivalent to solving the Generalized Chinese Remainder Theorem problem for a set of equivalencies, like this simple problem:

Find f(x) such that f(3)=2, f(-10)=1, f(6)=25.

is equivalent to the set of congruencies:

f(x)=2 mod (x-3)
f(x)=1 mod (x+10)
f(x)=25 mod (x-6)

This is now a Chinese Remainder Theorem problem, which is phrased in terms of polynomial division and remainders. This is a topic usually reserved for an Abstract Algebra class, so you probably wouldn’t bring it up in a high school setting. Perhaps you could mention it in passing, at least.

Solving this problem is of obvious value to any application requiring curve fitting. But over a finite field, it also has applications in cryptography.

All that being said–no real world application is really necessary at all. Like Bowen said, just discussing good problems that highlight algebraic structure is of value in and of itself.

In general, division of polynomials with clean answers isn’t going to make a whole lot of sense, because if the answer is a simple polynomial, it almost guarantees that the numerator was derived as a product to begin with…

Following from Bowen, you could have some problem where a cubic polynomial is given, and from a plot, you can see that it has a root at x = 1. Then you could use division to find the remaining roots (if real, or show that the remaining roots are complex).

@josh: I have seen it used for calculating running time of computer programs. (i.e. we know a program takes (5x^2+2x-1) / (2x-3) steps, but we only care about the largest polynomial term after division) That requires a LOT of excess background to explain what the problem is about, so I don’t know if I could write it in an Algebra-book friendly format. I’ll think about it though.

As for Algebra 1, I don’t know of a good single-author book and I’m not sure why there isn’t one. Key Press’ “Discovering Algebra” had the feel of the Serra book but I didn’t like the content much.

As usual I am ridiculously biased toward CME Project (the series I worked on). The series is much more focused on mathematical habits of mind like generalizing from repeated calculation than on Khan-style process work. Process matters too, but I feel that learning the “why” helps kids understand and remember the “how”.

Iambic pentameter entertains not just dogs but all sorts of animals! So do quadratics. Try throwing your dog a quadratic, he’ll bring it right back.

I can’t tell you how often I feel this way about the problems in our math books. Anything that feels contrived to us also feels contrived to our students, you can bet on it.

Now here’s the thing–even things that *aren’t* contrived, like projectile motion application problems, STILL don’t apply to the lives of our students. We can be sure that 99% of our students will never need to model projectile motion after they graduate from school. For the 1% who study math and science, these application problems will be important. But I can’t think of a single skill in a typical Algebra 1 curriculum that’s needed on a daily basis after graduation.

All that being said, it’s disingenuous to tell kids, “you are going to use this someday,” no matter how important the application may or may not be. Most likely, it’s NOT true that they will use it.

A better motivator is to remind students that they are being liberally educated–prepared for society, so they can talk intelligently with all kinds of people about all kinds of subjects. And it keeps they’re possibilities open, because some of them just might use it. Most of all, math is beautiful and fun and should always be treated as such. What student doesn’t love a good puzzle/problem that’s the perfect level of difficulty?

THIS hot air balloon problem, however, can go out with the trash.

Harold Jacobs: Geometry
Michael Serra: Discovering Geometry

@Bowen, thanks for the insight into how books are written. I have been curious about that process.

That brings me to my next question… I don’t know of any single author Algebra texts. Is Geometry a nice choice because it is a “new” system to the students? Because the author can build from the ground up? Do we not trust the students to be able to handle the abstract system of Algebra on the real numbers? Even for an Algebra 2 course? “Modern” algebra 1 and 2 texts are just “how-to” manuals, that sadly get thought of as “what math is” by students and the general public alike. Kahn Academy is the same thing.

Do english books get bogged down with such things? “you can use iambic pentameter to entertain your dog. Now lets drill.”

…

I suppose Foerster has an Algebra and Trigonometry book that is single author. I don’t have as much experience with it, though.

There are plenty of interesting problems to solve in the topic though. In dividing polynomials you can talk about what it means to divide and leave a remainder. Why is it that when I divide by (x-3), I can know that I’ll be able to get a constant remainder, and that the remainder when f(x) is divided by (x-3) is always f(3)? What is an acceptable remainder when dividing by (x^2 + 1)? How do the remainders when dividing by (x^2 + 1) relate to complex numbers?

There’s nothing real-world about it. But the analytic thinking (how is structure X like structure Y, and how are they different) is pretty universal.

The best I can come up with is the concept of Lagrange Interpolation (a method to fit a polynomial to given data) but that’s really more about the form and interpretation of expressions like 3(x-1)(x-6)(x+5) than anything to do with multiplying and dividing polynomials. Alright, multiplying maybe.

You can use Lagrange Interpolation to show that a question like “What number comes next: 1, 3, 5, ?” can be answered with any number at all, by finding a function that fits the sequence 1, 3, 5, 127 (for example). “Real world” that ain’t.

I guess then we can argue whether these are worthy topics for students given the relative lack of real-world application. I think the abstract nature of the topic makes it easier to focus on high-level thinking skills, and the understanding of things like quotient and remainder help students cement their understanding of the same concepts for numbers.