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Long Divison...Again
Posted:
Feb 29, 2008 12:09 PM
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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
P(x) = 12 x^5 - 7 x^4 - 11 x^3 + 28 x^2 + 13 x - 7
by
D(x) = 3 x^2 + 2 x - 1.
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>)
is
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
- -15 x^4 - 7 x^3 + 28 x^2 + 13 x - 7.
So
P(x) = [4 x^3 D(x)] - 15 x^4 - 7 x^3 + 28 x^2 + 13 x - 7.
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
P(x) = [4 x^3 D(x)] + [(-5 x^2) D(x)] + <still more junk>,
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
<http://clem.mscd.edu/%7Etalmanl>
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