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

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In summary, the conversation discusses the number of solutions for a polynomial equation and how it relates to rotating 2 pi in both directions. The Fundamental Theorem of Algebra states that every non-zero, single-variable, degree n polynomial with complex coefficients has n complex roots counted with multiplicity. This means that for a z^n equation, there will be n solutions. Additionally, the conversation touches on the idea that for a z^3 equation, there will be three distinct roots.
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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.
 

1. How do you determine the number of solutions for z?

To determine the number of solutions for z, you need to analyze the given equation or problem. Depending on the type of equation, you can use different methods such as substitution, elimination, or graphing to solve for z. The number of solutions will depend on the type of equation and the values of the variables.

2. Can an equation have more than one solution for z?

Yes, an equation can have more than one solution for z. This is known as a multiple solution or infinite solution. In this case, there are multiple values of z that satisfy the equation. This can happen when the equation has variables on both sides or when the equation is a quadratic with two solutions.

3. What does it mean if an equation has no solution for z?

If an equation has no solution for z, it means that there is no value of z that satisfies the equation. In other words, the equation has no solution when the solution set is empty. This can happen when the equation is inconsistent or when the values of the variables do not satisfy the equation.

4. How do you know if an equation has an infinite number of solutions for z?

An equation has an infinite number of solutions for z when the solution set is all real numbers. This means that any value of z will satisfy the equation. This can happen when the equation is an identity, where both sides of the equation are equal regardless of the value of z.

5. Can an equation have a complex solution for z?

Yes, an equation can have a complex solution for z. A complex solution is a solution that involves imaginary numbers, such as √-1. This can happen when the equation is a quadratic with no real solutions or when the equation involves complex numbers in the coefficients or constants.

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