Complex Numbers: 4th Degree Polynomial

[8daysAweek]

Homework Statement

Solve the following equation:

$$z^4+z^3+z^2+z+1 = 0$$

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:
$$(z+a)(z^3+bz^2+cz+1/a) = 0$$
I discovered that I need to solve 3rd degree polynomial just to do that.
$$a^3-a^2+1 = 0$$
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.

 

[Petek]

Try multiplying the original equation by z - 1.

 

[Hurkyl]

The lesson here is that you shouldn't forget what you learned in earlier classes. That polynomial is a geometric series, is it not?

 

[gomunkul51]

@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.