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Topic: [ap-calculus] Rational Zero Theorem Test
Replies: 2   Last Post: Sep 17, 2012 10:25 AM

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Dave L. Renfro

Posts: 2,165
Registered: 11/18/05
Re:[ap-calculus] Rational Zero Theorem Test
Posted: Sep 17, 2012 10:25 AM
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Michael Penigian wrote:

> Bob, thank you. I'm trying to gather up and make sense of
> all of these resources. There haven't been enough released
> multiple choice sections from the last few years, and there
> was an overhaul to the MC back in the late 90s I believe so
> some resources are outdated. Other resources such as 5 Steps
> to a 5 have difficult continuity questions involving the
> Rational Zero Test. I haven't seen it other than there,
> but would like to be safe. I see what you're saying about
> the main focus being on the Calculus and will try to keep
> that in mind throughout the year.

There is a middle position between fully using the rational
root theorem and not doing anything along these lines. Tell
the students about the "Factor Theorem" and use it in class
examples from time to time. This is the theorem that says
x = r is a root of the polynomial P(x) if and only if x - r
is a factor of the polynomial. The "if" half is the more
familiar result that students certainly know (maybe when
phrased differently), so it's the "only if" half I'm talking
about here.

Here are two examples. In each case, you can find the quotient
when P(x) is divided by x - r by using long division or by using
synthetic division.

1. Factor x^3 - 8.

Clearly, x^3 - 8 = 0 has x = 2 as a solution (no factoring
needed to find this solution), so x - 2 is a factor of x^3 - 8.

2. Factor x^5 + a^5.

Clearly, x^5 + a^5 = 0 has x = -a as a solution, so x + a
is a factor of x^5 + a^5.

The idea with this middle position is that you might be able
to easily get your hands on a root (by some method that doesn't
involve factoring, of course), and in these cases you can
immediately obtain a factorization (not necessarily a complete

For some more elaborate examples that involve polynomials
of more than one variable, for those wishing to become
connoisseurs of this method or for those who have students
that participate in math competitions, see the following

[ap-calculus] Teaching Factoring in Algebra II [2 December 2005]

I've also posted a couple of my old handouts on this method at

The shorter and more elementary handout (factor-th.pdf) was
designed for use in a standard U.S. HS/college algebra or
precalculus course. The longer and more advanced handout
(factor-th-more.pdf) was written as a supplement for a
first semester abstract algebra course that I've taught twice
(2002 and 2004).

Dave L. Renfro

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