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Roots and the Bisection Method


Date: 08/01/98 at 21:27:12
From: JCG
Subject: Polynomials

What is the Bisectional Method? How would I solve 2x^3+x^2+5x+2 = 0  
or any other polynomial? Can the method be used by something greater 
than the 3rd degree?


Date: 08/05/98 at 14:30:39
From: Doctor Benway
Subject: Re: Polynomials

The bisection method is one of my favorite surefire methods for finding 
roots of polynomials. The great thing about this method is that it 
works for everything, at least everything normal that you're likely to 
run into at this point in your mathematical education.

Have you ever watched the game show "The Price is Right"? That show 
actually uses a form of the bisection method, in a way. For example:

   Bob Barker: This lovely toaster oven/can opener from General 
               Electric can be yours if you can guess the price within 
               thirty seconds. Start the clock.
   Contestant: Ummm....$20
   Bob Barker: Higher
   Contestant: $100
   Bob Barker: Lower
   Contestant: $60
   Bob Barker: Lower
   Contestant: $40
   Bob Barker: Higher
   Contestant: $50
   Bob Barker: Correct!  You win!
   Contestant: Wheeeeeeeee!

As you notice, the contestant made two initial guesses, $20 and $100, 
which were too low and too high respectively. This told her that the 
price was between these two guesses. She then guessed halfway between, 
$60, and found that was too high, so it must be between $20 and $60.  
Her next guess, $40, was too low, so she knew it was between $40 and 
$60. She guessed $50, halfway between, and she was correct! 

This example turns out to be very similar to how the bisection method 
works. The usual method is to first plot a function to get an idea 
about where the roots are. Say f(x) has a root somewhere between 2 and 
3 when you graph it. Your goal is to guess numbers as above until you 
get a number that turns out to be zero when you plug it in. Your 
initial "guesses" are 2 and 3. To find out whether you need to go 
"higher" or "lower," you plug the numbers you are guessing into the 
function. In this case, f(x) being positive or negative will tell you 
whether you need to go higher or lower. The best way to do this is to
look at the graph to tell which way you should go. If you have an 
increasing function on that interval, a negative number means you 
guessed too low, and a positive number means you guessed too high (see 
below):  

                /   +
               /   too high
   ___________*__________
             /    
     -      /
    too    /
    low   /

If the function is decreasing, then the opposite rule applies. 

Where "bisection" comes in is that the best way to do this is to pick 
the midpoint of the interval you know the root to lie in, just as the 
contestant picked midway between the price range she knew. Eventually 
you will come closer and closer to the answer, and when you feel you 
have an x where f(x) is close enough to being zero, then you can put it 
down as an approximation for the root. 

This works for all continuous functions. It's not the best method, but 
it is certainly the easiest. If you take a class such as Numerical 
Analysis, you'll learn better methods for approximating roots.    

Hope this helps.

- Doctor Benway, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   
    
Associated Topics:
High School Polynomials

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