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Math Help - Complex Numbers and Cubic Polynomial Roots

  1. #1
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    Complex Numbers and Cubic Polynomial Roots

    Hello everyone,

    Let w be one of the three complex numbers having the property that
    w^3 = -4 + 4*sqrt(3)i.

    What is the |w|?

    Also if a = 2Re(w) how can one show that a is a root of the polynomial:
    p(z) = z^3 - 12z + 8.

    Thanks for your time.


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  2. #2
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    First note that \left| {z^3 } \right| = \sqrt {16 + 48}  = 8. So |z|=?

    I do not follow the second part of the question.
    Is it related to first part?
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  3. #3
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    Quote Originally Posted by JoAdams5000 View Post
    Hello everyone,

    Let w be one of the three complex numbers having the property that
    w^3 = -4 + 4*sqrt(3)i.

    What is the |w|?

    Also if a = 2Re(w) how can one show that a is a root of the polynomial:
    p(z) = z^3 - 12z + 8.

    Thanks for your time.


    Can you double-check your polynomial ?

    Should it be p(z)=z^3-12z+8\sqrt{3} ?

    EDIT: no, it isn't, I used a 60 degree angle instead of a 30 degree one!
    Last edited by Archie Meade; September 17th 2010 at 03:26 PM. Reason: took the wrong acute angle!
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  4. #4
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    Thanks!

    Part b is related to the first part, so

    a= 2*Re(w) where w is the be one of the three complex numbers having the property that w^3 = -4 + 4*sqrt(3)i.
    Then I must show that a is a root of the polynomial z^3 - 12*z + 8.

    And the polynomial is p(z) = z^3 - 12z + 8

    Thanks in advance.
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  5. #5
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    Quote Originally Posted by JoAdams5000 View Post
    Thanks!

    Part b is related to the first part, so

    a= 2*Re(w) where w is the be one of the three complex numbers having the property that w^3 = -4 + 4*sqrt(3)i.
    Then I must show that a is a root of the polynomial z^3 - 12*z + 8.

    And the polynomial is p(z) = z^3 - 12z + 8

    Thanks in advance.
    Note that:

    1. Using polar representation, a = 4 \cos \left( \frac{2 \pi}{9} \right).

    2. Using standard trig identities, \cos^3 (\theta) = \frac{3}{4} \cos (\theta) + \frac{1}{4} \cos (3 \theta).
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