Can someone explain why this equation has 6 roots? Doesn't

>>8945143
It's not a polynomial in z.

>>8945149
could you elaborate?

>>8945143
It's no longer a polynomial once you introduce the conjugate.
Polynomials only allow you to add together scalar multiples of powers of the same variable.

>>8945158
well then how can the answer be "z =..." and then 6 solutions follow. Can you explain how they arrive at that conclusion?

>>8945163
Because z determines both z and its conjugate...

>>8945168
Ok I understand that but then I should be getting z as something to the fifth (I think?) power for this to all add up, and I'm not getting that. Could you explain how you would solve this equation a bit more in detail? I really wanna understand how this works

>>8945176
change it to rational expression, you'll get 2z^5 in the numerator

>>8945180
thanks that helped, although I'm still stumped because I'm now getting 2e^(5*i*x)/r = 1 and I don't really know what to do with the r, I tried just setting doing 2(r^5)*e^(5*i*x) = r^6 but I get stuck here as well... really appreciate the help btw

Oi, brainlet,
let z be equal to a + bi
and, obviously, z* = a - bi

Do that out and use an equality of complex numbers to get two equations, in which you should find 6 answers. Obvious one is z = 0

>>8945176

What the other guy is trying to tell you is that a polynomial has to have, algebraically, the EXACT SAME THING in all the terms for the fundamental theorem of algebra to work as stated.

For example, x^4 - x^3 + 2x + 5 = 0 is a polynomial. The x is present in all terms and "follows the rules" of what a polynomial is. The degree of the polynomial is four (the term with the highest exponent), the coefficient of the quadratic term is zero, and in the last term, x is raised to the power of zero. Specifically, the above is a /univariate polynomial equation/. This is the fancy and technically correct way of saying that the above thing has ONE VARIABLE, involves a POLYNOMIAL, and is an EQUATION where one side is a polynomial, and the other side is set equal to zero. Let me call these things UPEs for short. x^2 - y = 0 is NOT a UPE (two variables). x^3 - x is NOT A UPE (not an equation, just a polynomial, an expression). ax^2 + bx + c = 0 IS a UPE, where a-c are the appropriate coefficients. The thing in the OP is NOT a UPE (two algebraically distinct variables, though in specific cases they may evaluate to the same thing).

Statements about "polynomials" like the quadratic formula, the cubic formula, Abel-Ruffini and the fundamental theorem of algebra and so on are generally statements about /univariate polynomial equations/. Mathematicians just call them "polynomials" in general even though they're often /polynomial equations/, because the above noise is a pain in the ass to spit out, as you're gathering.

Let's be lazy and plug this into Wolfram Alpha. Take the below string, plug it in there, and see what WA tells you. Notice especially that there is a /difference/ between whether we assume z is real, or complex (real numbers have complex conjugates as well).

2 z^2 - conjugate(z)^3 = 0

>>8945143
Note that what you have there is actually a rational function. And if you have studied those functions, you know that what determines the roots of a rational function is the polynomial at its numerator.

Now, what degree does the numerator of this rational function have?

>>8945197
Shitty and unnecessarily convoluted approach, you are the brainlet here. Better one would be:

[math] z=Re^{i \theta} [/math]
[math] 2 R^2 e^{2 i \theta}=R^3 e^{-3 i \theta} [\math]
This form of the equation clearly shows R=z=0 is a solution
[math] 2=R e^{-5 i \theta} [\math]
R must be 2 since the modulus of the exponential is 1 and R>0 by definition.
To get a strictly real value of [math]Re^{-5 i \theta}[/math] we need [math] -5 \theta = 2 \pi n, n=0,1,2,... [/math] which is satisfied for [math] \theta=0, 2\pi/5, 4\pi/5, 6\pi/5, 8\pi/5 [\math]

>>8945213
>*teleports behind you*
>unnecessarily convoluted approach, kiddo
>*kills you with shit formatting*

>>8945213
Wow your dad should have pulled out

>>8945223

either that or his mother should have swallowed him, same diff :^)

>>8945222
>heh... nothing personnel, piggot (that's a portmanteau of "pig" and "faggot" -- it means pig-faggot)