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Math Help - a complex numbers problem

  1. #1
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    a complex numbers problem

    Show that all roots of the eq. (z + 1)^3 + z^3 = 0 lie on the line
    x = -1/2
    thank you.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by unicorn
    Show that all roots of the eq. (z + 1)^3 + z^3 = 0 lie on the line
    x = -1/2
    thank you.
    First observe that a cubic has at least one real root. So if all the roots
    lie on x=-1/2 then z=-1/2 should be a real root.

    This checks out, so we know that z+1/2 is a factor of the cubic,
    and so we use long division to find the remaining quadratic factor:

    <br />
 (z + 1)^3 + z^3=2 (z+1/2)(z^2 + z + 1)=0<br />

    Now we can use that quadratic formula to verify that the two remaining
    roots (which are the roots of z^2 + z + 1) have real parts -1/2
    (which they do).

    RonL
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  3. #3
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    thanks
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