Complex Numbers: 4th Degree Polynomial

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Homework Help Overview

The discussion revolves around solving the equation z^4 + z^3 + z^2 + z + 1 = 0, where z is a complex number. Participants explore methods to factor or simplify the polynomial, considering its properties.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to factor the polynomial into a product of a first-degree and a third-degree polynomial, expressing frustration with the complexity involved. Some participants suggest alternative approaches, such as multiplying by z - 1 or recognizing the polynomial as a geometric series.

Discussion Status

Participants are actively engaging with the problem, offering different perspectives and methods. There is a recognition of the polynomial's geometric series nature, which may lead to further exploration of its roots. No consensus has been reached yet.

Contextual Notes

There is an indication that the original poster feels the problem may be more complex than expected for a homework exercise, suggesting a potential mismatch between the problem's difficulty and the homework context.

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Homework Statement



Solve the following equation:

[tex]z^4+z^3+z^2+z+1 = 0[/tex]

z is a complex number.

2. The attempt at a solution
I was trying to factorize it to 1st degree polynomial multiplied by 3rd degree polynomial:
[tex](z+a)(z^3+bz^2+cz+1/a) = 0[/tex]
I discovered that I need to solve 3rd degree polynomial just to do that.
[tex]a^3-a^2+1 = 0[/tex]
This is too much mess for a small homework exercise. I think that there is a technique that I am not aware of.




Thank You.
 
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Try multiplying the original equation by z - 1.
 
The lesson here is that you shouldn't forget what you learned in earlier classes. That polynomial is a geometric series, is it not?
 
@Hurkyl Very perceptive of you ! :)

Through the use of the finite geometric series sum you can find all the roots !

@Petek it's the same as multiplying by (1-z), you right.
 

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