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Math Help - Imaginary roots

  1. #1
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    Imaginary roots

    Let P(z) = z^3 + az^2 + bz + c, where a, b and c E R. Two of the roots of P(z) = 0 are -2 and (-3+2i). Find the value of a, b and c

    Thanks in Advance,
    Adam
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    Quote Originally Posted by Adam We View Post
    Let P(z) = z^3 + az^2 + bz + c, where a, b and c E R. Two of the roots of P(z) = 0 are -2 and (-3+2i). Find the value of a, b and c

    Thanks in Advance,
    Adam
    You might want to let z = x + iy so it's easier to distinguish between the real and complex parts.

    To find the third root, remember that if a complex number is a root, then its conjugate is also a root.

    Then P(Z) will equal the product of its factors, and (remembering to match real with real, imaginary with imaginary!) solve for a,b,c.
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  3. #3
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    Quote Originally Posted by Unenlightened View Post
    You might want to let z = x + iy so it's easier to distinguish between the real and complex parts.

    To find the third root, remember that if a complex number is a root, then its conjugate is also a root. Mr F adds: Provided the coefficients of the polynomial are all real (which they are in this case).

    Then P(Z) will equal the product of its factors, and (remembering to match real with real, imaginary with imaginary!) solve for a,b,c.
    And note that a fast way to expand (z - (\alpha + i \beta))(z - (\alpha - i \beta)) is to note that it's equal to (z - \alpha - i \beta)(z - \alpha + i \beta) = ([z - \alpha] - i \beta)([z - \alpha] + i \beta) = (z - \alpha)^2 + b^2 = \, ....
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