How do you know how many solutions there are for z?

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The discussion centers on determining the number of solutions for the variable z in polynomial equations. It highlights that a polynomial of degree n will have exactly n complex roots, as stated by the Fundamental Theorem of Algebra. The participants explore the specific case of z^3, noting that it yields three distinct solutions due to the nature of cubic equations. They also question whether the same logic applies to higher-degree polynomials, such as z^4, which would theoretically have four solutions. The conversation emphasizes the importance of understanding polynomial behavior and root multiplicity in solving such equations.
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Homework Statement
z^3+8i=0
Relevant Equations
r^3cis3t = 8cis(-pi/2)
Hi everyone

How do you know how many solutions z has
a) in this problem
b) in general?

I understand that they are rotating 2 pi from (-pi/2) in both directions to get the other two solutions. Should this be done in all problems?
Is it simply a coincidence that there are three solutions when z is in the third power?

If the problem needed to be solved for z^4, would there be four solutions? Or only three (again by rotating 2 pi in both directions)? Thanks
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... and since ##X^3+a## and ##(X^3+a)'=3X^2## have no common zeros if ##a\neq 0## we also know that there will be three pairwise distinct roots.
 

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