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Posted:
May 5, 1998 4:04 PM


OFFICE MEMO: Dick Pilgrim Date: Tue, 5 May 1998 13:02:44 0700 To: mathteach@forum.swarthmore.edu From: rpilgrim@ucsd.edu (Dick Pilgrim) Subject: re: Descartes' Rule of Signs
I was curious about the appearance of the topics of "theory of equations" in current College Algebra/Precalculus textbooks published for the college trade. I looked at 8 texts sent to me with copyrights >= 1997, ranging from a fifth edition of what one could call a "traditional" text to a new one touting "A Graphing Approach." Synthetic division was a topic in all 8 texts  some with more emphasis than others  in two cases as a shorthand way to do polynomial evaluation via a "nesting algorithm." Five of the texts had Descartes' Rule of Signs as a topic. There is no mention of the NCTM Standards in these texts nor is there mention of the AMATYC (Amer. Math. Assoc. of Two Year Colleges) Standards.
I used to introduce these as "tricks" in my honors calculus class in the 70s to speed up the computational drudgery in graphing polynomials. I had a class handout with the "proofs" for the interested students but we generally only spent a few hours with them. I do not see any good reason to do them today.
A similar question: As calculators like the TI92 become more common, what about the time spent in elementary calculus with techniques of integration?
(I have eliminated most of the messages and replies for brevity. The full content may be found in the message shown in the heading below.)
Dick Pilgrim  * Date: Tue, 05 May 1998 13:22:32 0400 (EDT) * From: "Ralph A. Raimi" <rarm@db1.cc.rochester.edu> * Subject: Re: Descartes' Rule of Signs * * On Tue, 5 May 1998, Karl Casper wrote: * > The book being used is Glencoe Algebra 2. The chapter is 8. The * * > They are given the remainder and factor theorems (without proof) and * > shown that synthetic division of a polynomial f(x) by (x+2) is the * > same as f(2) (Surely, the use of a calculator to find f(2) is far * > better than learning synthetic division for this purpose). * * Here already is a puzzle: why *synthetic* division? The process * of dividing one polynomial by another, to get a quotient polynomial plus * the rational fraction whose numerator has smaller degree than the * denominator, is a valuable illustration of things that turn up in other * places in mathematics, even if it isn't obviously useful. Only when the * denominator is of form xa does the "Remainder theorem" produce the result * f(a) for the remainder. The process of dividing P(x) by Q(x) is not * difficult, and can be made to illuminate the "long division" algorithm for * decimal numbers used in schools since the Renaissance. Replacing this * process in the particularly simple case Q(x) = xa by *synthetic* division * is stupid. It must have been invented by some schoolteacher who got tired * of repeating all those powers of x, and so cut away some of the symbols. * That would be very useful for any person who can make a living dividing * polynomials by linear polynomials, but I haven't met such a person yet, * not even in the days before calculators. To teach it in schools today is * the sort of waste of time that conceals mathematics from the public. * CUT OUT BY DICK PILGRIM * > * > I regard that spending three weeks on these topics an incredible waste * > of time. If the NCTM standards lead to implementations of this kind, * > clarification is needed. * > * > Karl Casper * > Department of Physics * > Cleveland State University * > Cleveland, Ohio * > * > * > * > * * * Ralph A. Raimi (Mathematics) Tel. 716 275 4429, or (home) 716 244 9368 * University of Rochester Fax. 716 244 6631, or (home) 716 442 3339 * Rochester, NY 14627 Homepage: http://www.math.rochester.edu/u/rarm



