What is the p/q Method in Solving Quartic or Higher Order Polynomials?

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Discussion Overview

The discussion revolves around the p/q method in solving quartic or higher order polynomials, exploring its effectiveness and the general approach to solving such equations. Participants share various methods and insights related to polynomial equations, including specific examples and theoretical considerations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about solving a quartic equation, comparing it to a simpler quadratic equation.
  • Another participant suggests using substitution, providing an example where x^4 + x^2 = 0 is transformed into a quadratic equation in terms of y.
  • A participant notes that the full solution of a quartic equation is lengthy and provides a link to a solution instead of detailing the calculations.
  • It is mentioned that a general solution for polynomials of degree higher than quartic cannot be found using basic arithmetic operations and roots, referencing Galois theory.
  • A question is raised about the effectiveness of the p/q method for finding roots of quartics or higher order polynomials, with a request for a simple example.
  • Another participant asserts that the p/q method guarantees four exact analytic roots for quartic equations.

Areas of Agreement / Disagreement

Participants express differing views on the effectiveness and applicability of the p/q method, with some asserting its guaranteed success for quartics while others focus on the limitations of finding general solutions for higher order polynomials. The discussion remains unresolved regarding the overall effectiveness of the p/q method.

Contextual Notes

Participants reference various methods and theories without reaching a consensus on the best approach for solving quartic equations, highlighting the complexity and length of solutions involved.

Who May Find This Useful

Individuals interested in polynomial equations, particularly those studying higher order polynomials and their solutions, may find this discussion relevant.

newton1
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how to solve the x from x^4+x^3...=...??
i mean from the equation for x power of 4
like the x^2-2x+1=0
the solve are x=1,x=1
 
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substitution

eg. x^4 + x^2 = 0
now y=x^2
--> y^2 + y = 0

you can solve this for y (= x^2)
 
the full solution of the quartic equation is fairly long. so rather than retyping a lot of calculations, i will link you to a solution here
 
Cutely, it's provable that you can't find a general solution (using addition/subtraction, multiplation/division, and roots) for polynomials in any order higher than quartic. The proof uses the same technique (Galois theory, loosely) as the geometric nom-constructibility proofs for things like trisecting an angle.
 
How effective is the p/q method for finding roots of quartics or higher order polynomials? Remind me how it works for a simple example.
 
Originally posted by Loren Booda
How effective is the p/q method for finding roots of quartics or higher order polynomials? Remind me how it works for a simple example.

what is the p/q method?

the quartic solution is 100%. it is guaranteed to give you four exact analytic roots to your quartic.
 

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