How to get 4 roots for z^4 +16 =0?

  • Thread starter Thread starter dla
  • Start date Start date
  • Tags Tags
    Roots
Click For Summary

Homework Help Overview

The problem involves finding the four roots of the equation z^4 + 16 = 0, which is situated in the context of complex numbers and polynomial equations.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to solve the equation by taking the square root of both sides, leading to z^2 = ±4i, but expresses uncertainty about the next steps. Other participants suggest factoring the equation and rewriting it in polar form, indicating different methods to approach the problem.

Discussion Status

Participants are exploring various methods to tackle the problem, including factoring and using polar coordinates. There is no explicit consensus on a single approach, but multiple lines of reasoning are being discussed.

Contextual Notes

Some participants are considering the implications of using complex numbers and polar forms, while the original poster is navigating the transition from z^2 to z.

dla
Messages
27
Reaction score
0

Homework Statement


Solve for z^4 +16=0

Homework Equations


The Attempt at a Solution


What I first did was square rooted both sides to get z^2 = ±4i, but I don't how to continue from there. I'm guessing we have to find the roots from z^2=4i and then from z^2=-4i separately any help will be much appreciated!
 
Physics news on Phys.org
try this first;

[tex]z^4+16=0[/tex]

[tex](z^2+4i)(z^2-4i)=0[/tex]
 
You could try rewriting the RHS of ##z^4 = -16## in Euler form, which may make the problem less fiddly.
 
In fact, in polar form, i= e^{i\pi/2} so [itex]\sqrt{i}= e^{i\pi/4}= \sqrt{2}/2+ i\sqrt{2}/2[/itex] and [itex]e^{-i\pi/4}= \sqrt{2}/2- i\sqrt{2}{2}[/itex]

If [itex]z^2= -4i[/itex] then [itex]z= \pm i(2)\sqrt{i}[/itex].
 

Similar threads

  • · Replies 22 ·
Replies
22
Views
2K
  • · Replies 8 ·
Replies
8
Views
2K
  • · Replies 7 ·
Replies
7
Views
2K
  • · Replies 11 ·
Replies
11
Views
2K
Replies
4
Views
2K
  • · Replies 8 ·
Replies
8
Views
2K
  • · Replies 7 ·
Replies
7
Views
2K
  • · Replies 2 ·
Replies
2
Views
1K
Replies
5
Views
2K
  • · Replies 6 ·
Replies
6
Views
2K