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Math Help - Problem Involving Polynomials and Their Roots

  1. #1
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    Problem Involving Polynomials and Their Roots

    If I need to find the a polynomial with real coefficients that possesses the roots -2 and 2-3i, I assume that I must do the following ...

    H(x) = (x + 2) (x - (2 - 3i) (x - (2 + 3i)
    = (x + 2) ((x - 2) - 3i) ((x - 2)+ 3i)
    = (x + 2) ((x - 2)^2 - (3i)^2)
    = (x + 2) ((x^2 - 4x + 4) - 9i^2)
    = (x + 2) (x^2 - 4x + 4 - 9(-1))
    = (x + 2) (x^2 - 4x +13)
    = x^3 - 2x^2 + 5x + 26

    Is this correct or am I missing something? Any suggestions or advice would be greatly appreciated. Thanks in advance.
    Last edited by headless chickens; October 30th 2009 at 08:54 AM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by headless chickens View Post
    If I need to find the a polynomial with real coefficients that possesses the roots -2 and 2-3i, I assume that I must do the following ...

    H(x) = (x + 2) (x - (2 - 3i) (x - (2 + 3i)
    = (x + 2) ((x - 2) - 3i) ((x - 2)+ 3i)
    = (x + 2) ((x - 2)^2 - (3i)^2)
    = (x + 2) ((x^2 - 4x + 4) - 9i^2)
    = (x + 2) (x^2 - 4x + 4 - 9(-1))
    = (x + 2) (x^2 - 4x +13)
    = x^3 - 2x^2 + 5x + 26

    Is this correct or am I missing something? Any suggestions or advice would be greatly appreciated. Thanks in advance.
    This is correct, it is a polynomial with real coefficients with the given roots, any other polynomial with those roots and real coefficients will be of the form of a polynomial with real coefficients times this one.

    CB
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