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Polynomial Function and Exponential Growth


Date: 06/08/99 at 22:39:29
From: roland relevo
Subject: Polynomials

Dear Dr. Math,

I have a homework question for my alegbra class. Here goes:

Find a polynomial function with integer coefficients that has 3, 2, -4 
and -1 as zeroes. All I know is that it is fourth degree polynomial. 
Help!

Here's another one. A story problem:

A colony of bacteria of 175000 is discovered in a petri dish. The 
colony doubles in size every 2.45 seconds. Model the population 
using exponential growth. What is the population 20 seconds after it 
is discovered? How long will it take to reach 3.5 billion?

Where do I begin?

Sincerely,
Roland R.


Date: 06/09/99 at 10:16:21
From: Doctor Rob
Subject: Re: Polynomials

Thanks for writing to Ask Dr. Math!

For the first problem, use this fact: If r is a root of a polynomial, 
then x-r is a factor. That should give you four factors of the 
polynomial.

For your second question, in the time from 0 to t seconds there are 
t/2.45 units of 2.45 seconds each. After each one, you have multiplied 
the population by 2, so the formula for the number of bacteria should 
be

   175000 * 2^(t/2.45)

There's your start.

Another approach is to say that the formula for exponential growth is

   P = a * b^t

for some constants a and b. Then when t = 0, P = a = 175000, and when 
t = 2.45, P = 2*a, so b^2.45 = 2, and you can solve for b.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Equations, Graphs, Translations
High School Exponents

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