Monday, February 13, 2012

Two things about "Theorems about Zeros of Polynomial Functions"

I'm prepping for a lecture called "Theorems about Zeros of Polynomial Functions."  With the exception of the Fundamental Theorem of Algebra and a couple results from said theorem, the lecture should really be called "Stuff That Became Extinct with the Graphing Calculator but We Make You Learn It Anyway."

Two Observations:

First, our text states the Rational Zero Theorem as follows:

 Just out of curiosity...why not write it more like the following:
If f(x) is a polynomial with integer coefficients written in descending order and if p/q is a zero of f(x), then p is a factor of the constant term of f(x) and q is a factor of the leading coefficient.
Personally, I've stopped using the scary polynomial function notation all together in my College Algebra classes.

Second Observation:  Descartes' Rule of Signs.  There's no need for me to say anything, I'll just copy what I found on PurpleMath.com:
Descartes' Rule of Signs is a useful help for finding the zeroes of a polynomial, assuming that you don't have the graph to look at. This topic isn't so useful if you have access to a graphing calculator because, rather than having to do guess-n-check to find the zeroes (using the Rational Root Test, Descartes' Rule of Signs, synthetic division, and other tools), you can just look at the picture on the screen.
Bam.

This doesn't, however, change the fact that we still teach Descartes' Rule of Signs at my college.  Right now, I show my students the following: 


As you can see, with the exception of a few minor changes, I basically just copy what's in the text.  There's got to be a better way.  Please enlighten me.

3 comments:

  1. I was thinking the exact same thing today as I was talking about this to my classes, the day after conferences. I was tired and crabby, so I know that is part of it. A lot of this chapter is not too useful though. This is the one time where I talk about complex numbers, because really, they won't use them too much, unless they take complex analysis (unlikely). We graph in Calculus, but don't really refer back to this. We learn other important points on the graph. And when in life are they going to be asked to factor an x^5 polynomial with every term included? But I guess I'm still tired and crabby, and will enjoy teaching the nuances of graphing a polynomial while my students are distracted by taking state tests next week :)

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    Replies
    1. I know...super frustrating.

      I don't know what PreCalc text you use, but we use Larson. And this year, we decided to just scrap Ch. 2 (Polynomial and Rational Functions). I was so much happier with this arrangement! I didn't feel nearly as rushed. Our reasoning was that the Algebra II teachers hit these topics pretty hard, so we were just flying through this chapter anyway. We will probably revisit bits and pieces toward the end of the year, but it will be more through the lenses of calculus as opposed to algebra (studying asymptotes as limits, etc.).

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  2. We use Coburn for honors Precalculus. We used to used Larson for precalc, just switched to Finney DeMana for regular precalc. I cruise through the Polynomials pretty quickly, but save all of the rational stuff for limits at the end...it makes more sense to do with discontinuities, etc.

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