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Thread: Roots of polynomial equations

  1. #1
    Newbie Femto's Avatar
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    Roots of polynomial equations

    I was happily answering some questions from my further pure textbook earlier until I encountered the last question; forgive me in advance for my lack of understanding, I'm only a sixth form student at the moment and I don't have as much knowledge as most people on this site!

    Here's the question:

    The roots of the equation $\displaystyle x^3 + ax + b$ are $\displaystyle \alpha, \beta, \gamma$. Find the equation with roots $\displaystyle \frac{\beta}{\gamma} + \frac{\gamma}{\beta}, \frac{\gamma}{\alpha} + \frac{\alpha}{\gamma}, \frac{\alpha}{\beta} + \frac{\beta}{\alpha}$.

    Initially I tried to use substitution but that failed epicly. Overall, I'm really confused - please could somebody assist me?

    Also, on a side note, I'm new here! Just thought you'd like to know
    Last edited by Femto; Dec 26th 2010 at 03:04 PM.
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  2. #2
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    Since $\displaystyle \alpha, \ \beta, \ \mbox{and} \ \gamma$ are roots, $\displaystyle x^3+ax+b=(x-\alpha)(x-\beta)(x-\gamma)$

    Does this help?
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  3. #3
    Newbie Femto's Avatar
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    Quote Originally Posted by dwsmith View Post
    Since $\displaystyle \alpha, \ \beta, \ \mbox{and} \ \gamma$ are roots, $\displaystyle x^3+ax+b=(x-\alpha)(x-\beta)(x-\gamma)$

    Does this help?
    Yes I kind of understand, but how do I obtain another equation with the new fractional roots presented in the question?

    Would it perhaps make sense to write it out as this?

    $\displaystyle (x - (\frac{\beta}{\gamma} + \frac{\gamma}{\beta}))(x - (\frac{\gamma}{\alpha} + \frac{\alpha}{\gamma}))(x - (\frac{\alpha}{\beta} + \frac{\beta}{\alpha}))$

    Really sorry, I'm getting a tad confused.
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  4. #4
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    $\displaystyle \displaystyle \left(x-\left(\frac{\beta}{\gamma}+\frac{\gamma}{\beta}\ri ght)\right)(\cdots)(\cdots)$
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  5. #5
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    Quote Originally Posted by Femto View Post
    Yes I kind of understand, but how do I obtain the new equation with the new fractional roots presented in the question?

    Would it help to say:

    $\displaystyle (x - (\frac{\beta}{\gamma} + \frac{\gamma}{\beta}))(x - (\frac{\gamma}{\alpha} + \frac{\alpha}{\gamma}))(x - (\frac{\alpha}{\beta} + \frac{\beta}{\alpha}))$?
    Now multiply it out.

    I would leave it factored, but if you need it in x^3.... format, you need to multiply.
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  6. #6
    Newbie Femto's Avatar
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    Quote Originally Posted by dwsmith View Post
    Now multiply it out.

    I would leave it factored, but if you need it in x^3.... format, you need to multiply.
    Thanks; yikes this looks time consuming.
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  7. #7
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    $\displaystyle x^3+ax+b=(x-\alpha)(x-\beta)(x-\gamma)=x^3-\alpha x^2-\gamma x^2-\beta x^2+\alpha\beta x+\beta\gamma x+\alpha\gamma x-\alpha\beta\gamma$

    $\displaystyle b=-\alpha\beta\gamma$

    $\displaystyle ax=x\alpha\beta +x\beta\gamma+x\alpha\gamma$

    $\displaystyle 0x^2=-\alpha x^2-\gamma x^2-\beta x^2$
    Last edited by dwsmith; Dec 26th 2010 at 02:06 PM. Reason: missed an alpha gamma x and beta x^2
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  8. #8
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    Quote Originally Posted by Femto View Post
    Here's the question:
    The roots of the equation $\displaystyle x^3 + ax + b$ are $\displaystyle \alpha, \beta, \gamma$. Find the equation with roots $\displaystyle \frac{\beta}{\gamma} + \frac{\gamma}{\beta}, \frac{\gamma}{\alpha} + \frac{\alpha}{\gamma}, \frac{\alpha}{\beta} + \frac{\beta}{\alpha}$.
    From the given we know that $\displaystyle \alpha+\beta+\gamma=0$,
    $\displaystyle \alpha\beta+\alpha\gamma+\beta\gamma=a$ and $\displaystyle \alpha\beta\gamma=-b $.
    You can use those and multiply out what you have setup.
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