Find all complex numbers z such that z^4 = -4 Note: All solutions should be expressed in the form a+bi, where a and b are real numbers. Please use Simons favorite factoring trick thank you!
I'll use Sophie Germain's Identity for this problem and don't think it would be posssible to use the Simon's favorite factoring trick. The Simon's favorite factoring trick is only applicable in:
- Two variable problems
- Degree of no more than 2.
z^4 + 4(1^4) = (z^2 + 2z + 2)(z^2 - 2z + 2)
You can find the complex roots by using the quadratic formula from there.
For this problem, you can even use Roots of Unity: https://artofproblemsolving.com/wiki/index.php/Roots_of_unity.