Roots of a 4th degree polynomial

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SUMMARY

The discussion focuses on finding the roots of the polynomial function 3x^4 - 960x^3 + 91500x^2 - 6272000x + 501760000. Users suggest various methods for root-finding, including Newton's Method and Ruffini's Method, while also considering the use of computational tools like Mathematica and MATLAB for efficiency. The consensus indicates that the roots are likely not rational, and the polynomial can be factored quickly using a computer. The conversation emphasizes the practicality of numerical methods over traditional algebraic approaches for this specific problem.

PREREQUISITES
  • Understanding of polynomial functions and their properties
  • Familiarity with Newton's Method for root approximation
  • Knowledge of Ruffini's Method for polynomial division
  • Basic skills in using computational tools like Mathematica or MATLAB
NEXT STEPS
  • Research Newton's Method for root-finding in detail
  • Explore Ruffini's Method and its applications in polynomial equations
  • Learn how to use Mathematica for polynomial root calculations
  • Investigate MATLAB's capabilities for numerical analysis and polynomial solving
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Mathematics students, educators, and anyone involved in computational mathematics or numerical analysis will benefit from this discussion, particularly those seeking efficient methods for solving polynomial equations.

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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