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Roots and the Bisection Method

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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?
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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
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/
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Associated Topics:
High School Polynomials

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