Solving Polynomials with Complex Coefficients

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Discussion Overview

The discussion revolves around methods for solving polynomials with complex coefficients, specifically focusing on finding factors and solutions. Participants explore the applicability of the factor theorem and other potential strategies for efficiently determining roots of such polynomials.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the efficiency of using the factor theorem to find the first factor of a polynomial with complex coefficients and wonders if there is a quicker method than trialing constant factors.
  • Another participant suggests that there may be an equivalent to the rational root theorem applicable to these problems and mentions the existence of a general closed form formula for cubic roots.
  • Some participants clarify the factor theorem, stating that if a polynomial P(z) has a root at a, then (z-a) is a factor, but they note that this does not provide guidance on what to guess as a root.
  • There is a discussion about whether the numerical solutions of certain polynomials correspond to the coefficients in a meaningful way or if it is merely coincidental.
  • One participant expresses uncertainty about the classification of the thread as homework-related, indicating that they are seeking different methods rather than needing help with homework.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the most efficient method for finding factors of polynomials with complex coefficients. There are differing views on the applicability and utility of the factor theorem and other methods discussed.

Contextual Notes

Participants express uncertainty regarding the existence of a quicker method for finding factors and the relationship between the coefficients and solutions of the polynomials. There is also a lack of clarity on how to classify the thread's content in relation to homework.

||spoon||
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Hey,

When solving polynomials over c that have complex coefficients such as:
z^3+(5i-4)z^2+(3-20i)z+15i
what is the easiest way to find your first factor? My textbook says to use the factor theorem, if you agree is there a quicker way to find a factor than by trialing the constants factors?

Also, for an equation such as:
Z^2+(3-4i)z-12i=0
or:
z^2-(3+2i)z+6i=0
the solutions are z=-3 z=4i and z=3 z=2i respectively. Is it just cpincidence that these solutions "correspond" to the same numerical values as is in the complex coefficient or is there some form of rule for this type of equation to quickly solve them at a glance?

Thanks

Btw i did not know wether this should go into honewirk or not. I chose not because it isnr homework and i can solve these with ease anyway lol. Just curious about different methods and speed :), cheers

-||spoon||
 
Last edited:
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||spoon|| said:
Hey,

When solving polynomials over c that have complex coefficients such as:
z^3+(5i-4)z^2+(3-20i)z+15i
what is the easiest way to find your first factor? My textbook says to use the factor theorem, if you agree is there a quicker way to find a factor than by trialing the constants factors?

Also, for an equation such as:
Z^2+(3-4i)z-12i=0
or:
z^2-(3+2i)z+6i=0
the solutions are z=-3 z=-4i and z=3 z=2i respectively. Is it just cpincidence that these solutions "correspond" to the same numerical values as is in the complex coefficient or is there some form of rule for this type of equation to quickly solve them at a glance?

Thanks

Btw i did not know wether this should go into honewirk or not. I chose not because it isnr homework and i can solve these with ease anyway lol. Just curious about different methods and speed :), cheers

-||spoon||

I'm not sure. I wonder if there is an equivalent to the rational root theorem for the types of problems you are trying to solve. I think there exists a general closed form formula for the roots of a cubic. For a more general treatment of algebraic solutions to polynomials pick up a book on group theory or modern algebra.
 
Last edited:
no I meant the factor theorem... If you have a polynomial P(z) such that P(a)=0 then. (z-a) is a solution.
 
||spoon|| said:
no I meant the factor theorem... If you have a polynomial P(z) such that P(a)=0 then. (z-a) is a solution.

Yeah, but that only tells you how to check if something is a root. It doesn't tell you what to guess.
 
sorry i misread your original thread, my bad. Very sorry.
 

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