Associated Topics || Dr. Math Home || Search Dr. Math

Factoring Polynomials

```
Date: 02/12/99 at 13:21:59
From: Joe Valentino
Subject: Factoring polynomials

I am an 8th grade math teacher. Some of my algebra students have asked
me how people would use the skill of factoring polynomials in a real
life situation. I am embarrassed to say that I am not really sure
where. Any suggestions?
```

```
Date: 03/22/99 at 12:08:03
From: Doctor Teeple
Subject: Re: factoring polynomials

Dear Joe,

Thanks for writing to Dr. Math.

Factoring is an important skill to have when you are trying to find
the zeros of a polynomial. For example, in mathematical biology, you
can model the change in populations by a system of differential
equations. In particular, I'm thinking of the Lotka-Volterra
equations, which model oscillating predator-prey systems. The
following is an excerpt from the textbook _Mathematical Models in
Biology_, by Leah Edelstein-Keshet. It's taken from Section 6.2, if
you want to look it up:

"In this section we explore a model for predator-prey interactions
that Volterra proposed to explain oscillations in fish populations in
the Mediterranean. To reconstruct his line of reasoning and arrive at
the equations independently, let us list some of the simplifying

1. Prey grow in an unlimited way when predators do not keep them
under control.
2. Predators depend on the presence of their prey to survive.
3. The rate of predation depends on the likelihood that a victim
is encountered by a predator.
4. The growth rate of the predator population is proportional to
food intake (rate of predation).

"Taking the simplest set of equations consistent with these
assumptions, Volterra wrote down the following model:

dx
--  = ax - bxy
dt

dy
-- = -cy + dxy
dt

where x and y represent prey and predator populations respectively;
the variables can represent, for example, biomass or population
densities of the species. ... "

Edelstein-Keshet then goes on the describe the terms of the equation:
a is the net growth rate of the prey in the absence of the predators,
c is the net death rate of the predators in the absence of prey, xy
approximates the likelihood that an encounter will take place between
predators and prey.

One thing to look for in a system of differential equations is the
steady-states of the system, which are the x and y that make both of
the differential equations 0. To find the steady-states, we set the
equations equal to 0 and factor to find the zeros:

dx
--  = ax - bxy = x(a - by)
dt

So either x = 0 or y = a/b.

If x = 0, then:

0 = -cy + dxy = -cy + 0   =>  y = 0

If y = a/b, then:

0 = -cy + dxy = -c(a/b) + d(a/b)x  => x = c/d

(0,0)  and (c/d, a/b)

With this information, we can ask whether the population will tend
toward or away from these steady-states. This is important to note
because we may want to know if the populations are tending toward
(0,0) and extinction.

Although this example is more about the applied part than the actual
factoring, note that we did start to find the steady-states by
factoring the first equation. Here, the factoring isn't particularly
difficult, but you can imagine that with more complex models the
factoring might be more complex. In general, factoring comes into play
when you want to find the zeros of equations.

I hope that you and your students enjoy this example. It's a pretty
well-known model in the mathematical biology world. Even if they
don't understand the details of a differential equations system, they
should be able to see where the factoring is important.

- Doctor Teeple, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculus

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search