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

  • Thread starter newton1
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In summary, the full solution of the quartic equation can be found using the p/q method, which is 100% effective in giving four exact analytic roots. However, it is provable that there is no general solution for polynomials of higher orders using traditional methods. The p/q method involves converting the equation to a simpler form by substituting a new variable and then solving for that variable, which can then be used to find the original roots.
  • #1
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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  • #2
substitution

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

you can solve this for y (= x^2)
 
  • #3
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
 
  • #4
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.
 
  • #5
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.
 
  • #6
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.
 

1. How do I solve x^4 if there is no equal sign?

To solve an equation with x^4, you need to set it equal to a number. For example, if the equation is x^4 = 16, you can solve by taking the fourth root of both sides, which gives x = ±2.

2. What is the process for solving x^4 equations?

The process for solving x^4 equations is similar to solving other equations with exponents. First, set the equation equal to a number. Then, take the appropriate root of both sides (in this case, the fourth root). Finally, solve for x by using any necessary algebraic techniques.

3. Can I use the quadratic formula to solve x^4 equations?

No, the quadratic formula is used to solve equations with x^2 as the highest power. For x^4 equations, you would need to take the fourth root of both sides and then solve for x using algebraic techniques.

4. What if the equation has a coefficient in front of x^4?

If the equation is in the form ax^4 = b, you can simply divide both sides by a before taking the fourth root. This will give you the solution for x without the coefficient.

5. Are there any special cases when solving x^4 equations?

One special case is when the equation has a negative exponent, such as x^-4. In this case, you would need to take the fourth root of both sides and then take the reciprocal to get the solution for x.

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