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

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The p/q method is a technique used to find roots of quartic and higher-order polynomials. It involves substituting variables to simplify the equation, as demonstrated with the example of x^4 + x^2 = 0, which can be transformed into a quadratic form. While the quartic equation can be solved analytically, it is noted that no general solution exists for polynomials of degree five or higher using basic arithmetic operations and roots. The p/q method is effective for quartics, guaranteeing four exact roots. This method's reliability contrasts with the limitations faced in solving higher-degree polynomials.
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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.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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