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Math Help - Complex Numbers - Finding Roots

  1. #1
    Member classicstrings's Avatar
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    Complex Numbers - Finding Roots

    Hi! I'm stuck on this question!

    (a) For β = 1 - i√3, write the product z - β and z - conjugateβ as a quadratic expression in z, with real coefficients.

    I got z^2 - 2z + 4

    (b) (i) Express β in mod-arg form. Ans: 2cis(-π/3)
    (ii) Find β^2 and β^3. Ans: 4cis(-2π/3) and 8cis(π)
    (iii) Hence show that β is a root of z^3 - z^2 + 2z + 4 = 0. Stuck on this part, "hence"?

    (c) Let the three roots be a, b, c. Let a be the point in the 1st quadrant, B the point on the real axis. Let c be the other root.

    (i) Find the lengths AB and CB.
    (ii) Describe the triangle ABC
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by classicstrings View Post
    Hi! I'm stuck on this question!

    (a) For β = 1 - i√3, write the product z - β and z - conjugateβ as a quadratic expression in z, with real coefficients.

    I got z^2 - 2z + 4

    (b) (i) Express β in mod-arg form. Ans: 2cis(-π/3)
    (ii) Find β^2 and β^3. Ans: 4cis(-2π/3) and 8cis(π)
    (iii) Hence show that β is a root of z^3 - z^2 + 2z + 4 = 0. Stuck on this part, "hence"?
    You have shown that (z - beta) (z - beta') = z^2 - 2z +4, so:

    (z+1)(z-beta)(z-beta') = (z+1)(z^2 - 2z +4) = z^3 - z^2 + 2z + 4

    Hence z=beta is a root of z^3 - z^2 + 2z + 4=0.

    RonL
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  3. #3
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    This PDF file can help you with this problem.
    Attached Files Attached Files
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by classicstrings View Post
    Hi! I'm stuck on this question!

    (a) For β = 1 - i√3, write the product z - β and z - conjugateβ as a quadratic expression in z, with real coefficients.

    I got z^2 - 2z + 4

    (b) (i) Express β in mod-arg form. Ans: 2cis(-π/3)
    (ii) Find β^2 and β^3. Ans: 4cis(-2π/3) and 8cis(π)
    (iii) Hence show that β is a root of z^3 - z^2 + 2z + 4 = 0. Stuck on this part, "hence"?

    (c) Let the three roots be a, b, c. Let a be the point in the 1st quadrant, B the point on the real axis. Let c be the other root.

    (i) Find the lengths AB and CB.
    (ii) Describe the triangle ABC
    Attached is a diagram of the position of the three points in the complex
    plane (the roots of z^3 - z^2 + 2z + 4 = 0), from which you should be able
    to answer (c) parts i and ii.

    RonL
    Attached Thumbnails Attached Thumbnails Complex Numbers - Finding Roots-gash.jpg  
    Last edited by CaptainBlack; April 24th 2007 at 02:07 AM.
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