Complex roots of cubic polynomial

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mathnerd2021

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One of the solutions of the equation z^3=w is Z1=-5√3+5i. What is one of the other two solutions if w is a complex number?

So one of the other roots will be a complex conjugate. It will be -5√3-5i. Multiplying we get 100.

How do we get the third root?
 
Dr.Peterson said:
Are you thinking they said w is real? This is a false conclusion.


What did you multiply? And why?


What have you learned about the roots of unity?
Click to expand...
The rule is that if the coefficients of a polynomial are real, then the complex roots come in conjugate pairs. The coefficient of z^3 is 1, which is real. So the conjugate root to the one already given will be -5√3-5i. Am I wrong?

I made a mistake in multiplying the roots. I needed to multiply the factors of the cubic polynomial.

So we have (z + 5√3 - 5i)(z +5√3 + 5i) (z + A) = z^3
I am not sure how to proceed from here.

The nth roots of unity are given by,
{\displaystyle \exp \left({\frac {2k\pi i}{n}}\right)=\cos {\frac {2k\pi }{n}}+i\sin {\frac {2k\pi }{n}},\qquad k=0,1,\dots ,n-1.}
 
mathnerd2021 said:
The rule is that if the coefficients of a polynomial are real, then the complex roots come in conjugate pairs. The coefficient of z^3 is 1, which is real. So the conjugate root to the one already given will be -5√3-5i. Am I wrong?
Click to expand...
But w is also a coefficient in the equation z^3 - w = 0, and they tell you it is complex. So, no, it isn't (z + 5√3 - 5i)(z +5√3 + 5i) (z + A) = z^3. You're barking up the wrong tree.
mathnerd2021 said:
The nth roots of unity are given by,
{\displaystyle \exp \left({\frac {2k\pi i}{n}}\right)=\cos {\frac {2k\pi }{n}}+i\sin {\frac {2k\pi }{n}},\qquad k=0,1,\dots ,n-1.}
Click to expand...
So how are the cube roots of a complex number w related?
 
pka said:
I find this post confused in the extreme. Is it possible for you to post the question exactly as it was given to you?
Click to expand...
One of the solutions of the equation z^3=w is Z1=-5√3+5i. What are the other two solutions if w is a complex number?
 
mathnerd2021 said:
One of the solutions of the equation z^3=w is Z1=-5√3+5i. What are the other two solutions if w is a complex number?
Click to expand...
We are being told that z1 is one of the three complex cube roots of the complex number w. Have you learned that the three roots are multiples of the given one, by the complex cube roots of 1?
 
No I've not learned this. Could you please point me to some websites that could help me solve this?
 
I'm very curious: What have you been taught, that they expect you to use? There should be something there we can start with.

Have you learned what is taught here? (It doesn't have an example like your problem, but you can use the ideas.)

My suggestion seems to me to be the only obvious way to solve your problem; though another way would be to cube z1 to find w, and then factor the equation over the complex numbers.
 
If you haven't learnt that then here's another way to approach the problem.

You can calculate w by substituting z1 in for z because you know that z^3 = w.
Then use your usual way to find the cube roots of your known w.
 
Harry_the_cat said:
If you haven't learnt that then here's another way to approach the problem.

You can calculate w by substituting z1 in for z because you know that z^3 = w.
Then use your usual way to find the cube roots of your known w.
Click to expand...
Thank you. I will try to work it out this way.
 
Dr.Peterson said:
I'm very curious: What have you been taught, that they expect you to use? There should be something there we can start with.

Have you learned what is taught here? (It doesn't have an example like your problem, but you can use the ideas.)

My suggestion seems to me to be the only obvious way to solve your problem; though another way would be to cube z1 to find w, and then factor the equation over the complex numbers.
Click to expand...
Thank you. I have done my undergrad in Math but I'm mostly out of touch with concepts after grade 11. However, I don't remember learning this topic ever. Is this a master's topic?
 
mathnerd2021 said:
Thank you. I have done my undergrad in Math but I'm mostly out of touch with concepts after grade 11. However, I don't remember learning this topic ever. Is this a master's topic?
Click to expand...
It's rather basic to the study of complex numbers. The nth roots of a number are spaced 2pi/n radians around the origin, so to find more roots given one of them, you just rotate through that angle, which you do by multiplying by a root of unity. This particular problem is not a standard one, but it uses only those ideas.

But I'm not asking so much what you happen to recall, but about the context of the question. Is it from a textbook? If so, what have they just taught before giving this problem? If it's from some other source, what is it?

Having asked that, I just searched and found it on various question sites (with slightly varying wording). But if you found it there, then you will have seen the answer (found by several methods, of which mine is probably the easiest).
 
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