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 
assumptions he made:

   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

So the two steady-states are:

   (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.

If you need more information, please write back.

- Doctor Teeple, The Math Forum
  http://mathforum.org/dr.math/