Roots of a 4th degree polynomial

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Homework Help Overview

The discussion revolves around finding the roots of a fourth-degree polynomial, specifically the polynomial given by 3x^4 - 960x^3 + 91500x^2 - 6272000x + 501760000. Participants express frustration with the complexity of the problem and explore various methods for root finding.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss methods such as Ruffini's method and Newton's method for finding roots. Some question the feasibility of rational roots given the polynomial's coefficients and suggest using computational tools like Mathematica or Matlab. There is also a suggestion to define a root algebraically and explore that approach.

Discussion Status

The discussion is active, with participants sharing methods and expressing varying opinions on the nature of the roots. Some guidance has been offered regarding specific methods, and there is acknowledgment of the potential for computational assistance. However, there is no explicit consensus on the best approach yet.

Contextual Notes

Participants note the challenge posed by the large coefficients in the polynomial and the potential for roots not being rational. There is also mention of an inside joke related to approximations in the context of the Newton method.

Grantismo
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Hi eveveryone I was just hoping for some quick help on frustrating physics related math problem. I won't go into detail on the actual problem becasue i know i found the correct polynomial but i was wondering if there was any easy way to find the roots to this polynomial:

3x^4-960x^3+91500x^2-6272000x+501760000=f(x)
*sorry i haven't figured out how to use latex or w/e it's called*rational roots seems rather arduous with the numbers involved. Any suggestions?(I know there is only one answer about 125.98 i think but i was wondering if there was a way to find an exact answer algebraically or with calculus or something)
 
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Apart from using Ruffini's method (rational roots) or Newton's method, both which will require time to yield answers, maybe since this isn't a mathematical problem you can use Mathematica, Matlab, etc... for your solutions.

Also you could have tried Descartes' sign rule, but that wouldn't have helped much anyway.
 
rational roots seems rather arduous with the numbers involved.
501760000 isn't a very big number. A computer should be able to factor that before you can blink. It can probably plug every number dividing 501760000 into that polynomial roughly as quickly.


i was wondering if there was a way to find an exact answer algebraically
Is there any reason why you can't simply define r to be a root of that polynomial, and then work with r?
 
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I don't think that the roots are rational now that I've looked at a it or a while,
Cyclovenom if you could explain any of those methods i might try them.
 
Grantismo said:
I don't think that the roots are rational now that I've looked at a it or a while,
Cyclovenom if you could explain any of those methods i might try them.

Certainly, i will try to answer any questions about the methods, but they are explained in these sites:

Newton's Method

http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/approx/Newton.html"

http://www.sosmath.com/calculus/diff/der07/der07.html"

Ruffini's Method

"[URL
 
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wow, the Newton method is PERFECT for what I wanted, plus it will also give my teacher a huge laugh (inside joke about approximations)
Thank you SOO much :)
 

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