Nonreal Roots

Date: 9/26/96 at 1:2:5
From: Anonymous
Subject: Re: Nonreal roots of Systems of Equations

I am teaching a precalculus class. Our text has a section on nonreal
roots of systems of equations and I am trying to figure out a good
way to present what it means. How would you do it? Picture it as
a 3d graph with i as the z axis? What would be a use for knowing
nonreal roots?

Thanks!

Diane Thompson

Date: 10/1/96 at 18:4:10
From: Doctor Ceeks
Subject: Re: Nonreal roots of Systems of Equations

Hi,

What kind of answer are you looking for?

I see it like this:  Polynomials factor completely into linear
factors and quadratic factors over the real numbers.  Sometimes the
quadratic factors do not factor into linear factors of the reals,
but if you extend the numbers to the complexes, then they do.
In other words, non-real roots imply that your polynomial cannot
be factored into linear factors over the reals...you will inevitably
have quadratic factors.

Factoring is a useful concept, so knowing how a polynomial factors
is very useful.

As a concrete example, since x^2+1 does not have real roots, we know
it cannot be written as a product of linear factors over the reals... 
it is "irreducible" over the reals.

If you aren't satisfied with this answer, please ask again, but do
try to explain what kind of answer you are looking for.

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/