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Math Help - Repeated factors

  1. #1
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    Repeated factors

    Find the value of a for which the function 2x^3-ax^2-12x-7 has repeated factors.

    If \alpha is the repeated root, (x-\alpha) is the factor
    f(\alpha)\equiv 2\alpha^3-a\alpha^2-12\alpha-7=0 ___(1)
    f'(\alpha)\eqiuv 6\alpha^2-2a\alpha-12=0 ___(2)

    \alpha (2)-3\times (1)\longrightarrow (-2a\alpha^2-12\alpha)-(-3a\alpha^2-36\alpha - 7)=0
    a\alpha^2+24\alpha+7=0
    This is as far as I got.
    Thanks!
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  2. #2
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    Quote Originally Posted by arze View Post
    Find the value of a for which the function 2x^3-ax^2-12x-7 has repeated factors.

    If \alpha is the repeated root, (x-\alpha) is the factor
    f(\alpha)\equiv 2\alpha^3-a\alpha^2-12\alpha-7=0 ___(1)
    f'(\alpha)\eqiuv 6\alpha^2-2a\alpha-12=0 ___(2)

    \alpha (2)-3\times (1)\longrightarrow (-2a\alpha^2-12\alpha)-(-3a\alpha^2-36\alpha - 7)=0
    a\alpha^2+24\alpha+7=0
    This is as far as I got.
    Thanks!
    That is a good way to start the problem. To continue, divide your equation (2) by 2, then multiply it by  \alpha and subtract the resulting equation from equation (1). That will get rid of the terms containing  a, leaving you with a cubic equation for  \alpha (for which you ought to be able to spot one of the roots).
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