Solving a Complex Polynomial Equation

[wam_mi]

Homework Statement

Happy New Year, everybody!

Problem: Solve (z^4)+1 = 0

Homework Equations

The Attempt at a Solution

Attempt: (z^4)+1 = 0
((z^2)+i)*((z^2)-i) = 0

Could anyone help me to complete this question please?

Many Thanks!

 

[Hootenanny]

HINT: Let $z = e^{i\theta}$

 

[HallsofIvy]

Homework Statement

Happy New Year, everybody!

Problem: Solve (z^4)+1 = 0

Homework Equations

The Attempt at a Solution

Attempt: (z^4)+1 = 0
((z^2)+i)*((z^2)-i) = 0

Could anyone help me to complete this question please?

Many Thanks!

So you know that z^2+ i= 0 and z^2- i= 0. Now you need to solve z^2= i and z^2= -i.

You can, as Hootenanny suggested, write i and -i in "polar form" and use the fact that
$$\left(re^{i\theta}\right)^{1/2}= r^{1/2} e^{i\theta/2}$$

 

[wam_mi]

So you know that z^2+ i= 0 and z^2- i= 0. Now you need to solve z^2= i and z^2= -i.

You can, as Hootenanny suggested, write i and -i in "polar form" and use the fact that
$$\left(re^{i\theta}\right)^{1/2}= r^{1/2} e^{i\theta/2}$$

Thanks a lot for your help HallsofIvy. I think I can do it now, thank you!

 

[wam_mi]

So you know that z^2+ i= 0 and z^2- i= 0. Now you need to solve z^2= i and z^2= -i.

You can, as Hootenanny suggested, write i and -i in "polar form" and use the fact that
$$\left(re^{i\theta}\right)^{1/2}= r^{1/2} e^{i\theta/2}$$

Thanks for the hint!