Quartic with complex coefficients

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SUMMARY

The discussion focuses on solving a fourth-order polynomial of the form x^4 + Ax^3 + (B_1 + B_2p)x^2 - (C + Ap)x + D + Ep = 0, where A, B_1, B_2, C, D, E are real parameters and p is a complex parameter. The user initially struggled with the application of Ferrari's method for complex coefficients but later confirmed its applicability. The primary inquiry revolves around classifying the roots based on the parameters, particularly whether the imaginary part of p influences the nature of the roots, suggesting that if Im(p) is not equal to zero, all solutions may be complex.

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  • Understanding of polynomial equations, specifically fourth-order polynomials.
  • Familiarity with Ferrari's method for solving polynomial equations.
  • Knowledge of complex numbers and their properties.
  • Basic concepts of root classification in polynomials.
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  • Research the application of Ferrari's method for complex coefficients.
  • Study the classification of polynomial roots based on parameters, focusing on complex variables.
  • Explore the implications of the imaginary part of complex parameters on polynomial solutions.
  • Investigate numerical methods for solving polynomials with complex coefficients.
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Mathematicians, engineers, and researchers dealing with complex polynomial equations, particularly those interested in root classification and numerical methods for solving higher-order polynomials.

ab959
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I am trying to solve a fourth order polynomial which is in the following form

x^4+Ax^3+(B_1+B_2p)x^2-(C+Ap)x+D+Ep=0

Where A, B_1, B_2, C, D, E, are real parameters and p is a complex parameter.

I have investigated many ways of solving this equation however there does not seem to be much information regarding complex coefficients. My solution is very messy for real coefficients but it still exists and I derived it using Ferrari's method. I am not sure if I can use this method in the case where p is complex.

Any suggestions will be much appreciated.
 
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Yes, Ferrari's method works for both real and comples coefficients.
 
OK great thanks I realized this soon after I posted. What I am more interested in knowing is if you classify the roots in terms of the parameters. I.e. knowing when there will be 4 real roots or complex roots etc. My intuition tells me however that if Im(p) not equal to zero then all solutions will be complex. This is possible for real coefficients however I don't know if this can be done for complex coefficients.
 

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