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Solving a Polynomial of Any Degree

Date: 06/02/2003 at 14:43:27
From: David
Subject: What method must be used to solve a polynomial of any degree?

How to solve a polynomial of any degree? I want to make a computer
program that is capable of solving any polynomial.

Newton's method can solve many polynomials, but it does not always 
give correct answers: sometimes the answers oscillate and at other 
times it might never converge. Is there a better way of solving 
polynomials?


Date: 06/04/2003 at 08:55:58
From: Doctor George
Subject: Re: What method must be used to solve a polynomial of any 
degree?

Hi David,

Thanks for writing to Doctor Math.

Using Newton's method successfully can be a challenge. There are 
some very complex ways to solve your problem. A simpler way is to 
combine Newton's method with the Bisection method.

Let's say that you know that f(a) < 0 and f(b) > 0. There is 
therefore a root of f(x) in the interval (a,b). Now follow this 
procedure.

1. Make a guess x_0 in (a,b) and compute x_1 using Newton's method.
2. If x_1 is in (a,b) then continue with Newton's method.
3. If x_1 is not in (a,b), then set x_1 = (a+b)/2. If f(x_1) < 0 the
   new interval is (x_1,b). Otherwise the new interval is (a,x_1).
4. Now go back to step 1 using x_1 in place of x_0 to get x_2, and
   so on until you converge.

Let's say that you find that 'c' is a root. Then there is a g(x) 
such that

         f(x) = (x-c)g(x)

         g(x) = f(x)/(x-c)

Now do Newton's method on g(x) to find the next root of f(x).

Does this make sense? Write again if you need more help.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Polynomials

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