Factoring PolynomialsDate: 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/ |
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