## Thursday, February 28, 2013

### Math problems of the week: polynomial division in UCMP vs. Wentworth's Algebra

I. The only problem in the entire University of Chicago Math Project Algebra involving the division of a multinomial by a polynomial (p. 730):

There are no problems in University of Chicago Math Project Algebra that involve polynomial long division.

II. The first problem set in Wentworth's New School Algebra involving polynomial long division (pp. 70-71) [click to enlarge]:

My 6th grade (home-schooled) daughter's work on the 38th of these problems (she did this problem just yesterday, and, though she's no math genius, needed no help from me):

III. Extra Credit

Polynomial long division provides lots of practice with the distributive law: here, one is constantly distributing both monomials and the negative operator across polynomials. In addition, polynomial long division shows you place value at a more abstract level: here, one can discuss, for example, the c2s and the cs places. It also shows you the long division algorithm playing out at a more abstract level--which I recall finding quite enlightening back when I did polynomial long division (a few decades before Reform Algebra eliminated it).

1. Is there a good reason not to have today's algebra students do lots of polynomial long division?

2. Is there a good reason to wait until p. 730 of an algebra text before doing any binomial by monomial division?

Anonymous said...

Should jr high or high school students spend lots of time doing polynomial long division? I have done lots of math but I have hardly ever divided one polynomial by another, so I'm leaning towards ... maybe not. The pertinent question is this: if you know how to divide polynomials, what else does that allow you to do? One obvious thing is if you want to factor a polynomial P(x) and you know one root, say r, then dividing by x-r gives you a lower degree polynomial and then you can try to factor that one [P(x) = (x-r)*Q(x)]. What else? If you want to simplify rational functions ... but is that really high on the list of things to do in Algebra I?

There is one thing that I like about this topic. When I learned to do it we did it "in-line" by writing P(x) = (x-r)*( ...) and then adding in and subtracting out terms as necessary to make it work. This is not as efficient as long division but it teaches the basic all-important algebraic technique: write down what you want to have and then subtract or divide out as necessary to make the left side equal to the right side.

Auntie Ann said...

I've been working through this section of the book the last couple of days, and it certainly teaches a number of important skills. It drives home working with exponents, keeping track of negatives, and is great practice for multiplying and distributing polynomials. Some things simply need to be practiced extensively to get them to sink in. This allows a student to practice multiple skills to mastery.

It's also fun; you have complicated problems that appear to be really hard, then you crank through them an get the right answer. I think teachers too often forget that working through difficult problems, and then proving you have the right answer is actually enjoyable. They see it too often as drudgery, instead of as accomplishment and achievement.