Two questionable assertions about long division have resurfaced recently:
1. Students need to study long division with integers before they can understand (or, "perform" if you're one of the list subscribers who seem to think that "perform" is a reasonable substitute for "understand") long division with polynomials.
2. For long division (especially?), the "How?" question naturally precedes the "Why?" question.
At best, these assertions result from unimaginative failure to think about alternate presentations. At second best, they are unthinking recitation of a party line that tries to rationalize its preconceptions without regard for underlying mathematics. At worst,... Well, I leave that to the imagination of the interested reader.
Polynomial division is, in fact, easier than division of integers. Both algorithms depend on place value, but the place value relevant to polynomial division is the place occupied by terms of a given degree---while the place value relevant to integer division is the place that designates power of ten. The difference that makes polynomial long division easier than integer long division is that polynomial calculations never require carrying or regrouping ("borrowing", to use a term some think archaic and others think proper). In polynomial long division, the next piece of the partial quotient never requires readjustment as it frequently does in integer long division. Thus, when we are asked to divide 3680 by 46, most of us can't reliably predict whether the first digit of the quotient is 7 or 8---or even 9. If we guess that it's 7 or 9 we must return and make a correction. That's because multiplying 7, 8, or 9 times 46 involves a carry whose effect is hard to foresee.
This effect disappears when we divide polynomials. In dividing A x^5 + B x^4 + ... by P x^2 + ..., the first term of the quotient is always (A/P) x^3. We never need to make a readjustment. [And let's note that in beginning algebra we don't expect students to do a long division to find A/P. Suggesting that we do is a quibble---not a substantive observation. The focus isn't and shouldn't be on that part of the algorithm; short division of simple integers is the most we need; writing the divison as the fraction A/P suffices, especially if we reduce the fraction.]
Some could find support in the foregoing analysis for deferring the introduction of the long division algorithm for decimal numbers until after the student has studied long division for polynomials. I'm not willing to go quite that far, but I have to admit that it's a strong case and I'm not altogether certain that the objections that come immediately to my mind are strong enough to counter that case.
As to whether "How?" naturally precedes "Why?" let's think about what we want when we are to divide the polynomial
We first conclude that the quotient, whatever it is, will have to have 4 x^3 as its leading term because multiplying (3 x^2 + 2 x - 1) by (4 x^3 + <Lower degree stuff>) is the only way to generate 12 x^5 as the highest degree term of the product. But the product
(3 x^2 + 2 x - 1) times (4 x^3 + <Lower degree stuff>)
12 x^5 + 8 x^4 - 4 x^3 + <other stuff>.
We want this last expression agree with P(x). The only way that's possible is (why?) for <other stuff> to be
The only way that the multiplication (4 x^3 + <junk>) times D(x) can generate that -15 x^4 term would be (why?) for <junk> to have (-5 x^2) as its leading term. But (-5 x^2) times D(x) is -15 x^4 - 10 x^3 + 5 x^2, which means that we now want to rewrite P(x) in the form
and we easily calculate that <still more junk> must be the cubic polynomial 3 x^3 + 23 x^2 + 13 x - 7.
That should be enough for the readers of this list to see where this goes. Further detail will, of course, be required for students making their first encounters with the ideas. The analysis develops the long divison algorithm for polynomials in detail---as it investigates (and answers!) the "Why?" question in the same detail. Neither question precedes the other.
The analysis above doesn't closely resemble the long division algorithm for polynomials as it is usually presented. In particular the efficient tableau is missing. It can come later, after the "Why?" and the "How?" question have both been addressed. That's exactly what every college algebra text I've ever seen does in its discussion of synthetic division by a monic degree-one binomial: Develop the efficient tableau from a more detailed calculation that explains where things come from. I refer the reader to such texts for an indication of what needs to be done. It is apparent from the universality of these presentations in traditional books that the argument that the approach is unsound is not one a traditionalist should take seriously. In fact, part of effective learning a good algorithm should consist of figuring out how to take something we understand and make it efficient.
Finally, let's remark that there is no reason why long division of integers shouldn't be presented in similar fashion. I worked with polynomials here because, as suggested earlier, the algorithm and its presentation are cleaner and easier, both practically and conceptually.
- --Lou Talman Department of Mathematical & Computer Sciences Metropolitan State College of Denver