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Math Help - 6th order Polynomial

  1. #1
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    6th order Polynomial

    Hello!

    I am looking for the roots of the following equation:

    -2.3704s^6-3.5555s^4-1.333s^2+1

    I was told to use the substitution x=s^2 but I still can't figure out how to do it. This has to be solved in an exam as part of a bigger question and should not take longer than a minute, so I can't go for other approches.

    Thanks,

    Uli
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  2. #2
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    May 2009
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    Did the original equation have rational coefficients, possibly

    -\frac{64}{27} x^3 - \frac{32}{9} x^2 - \frac{4}{3} x +1 = 0 ?

    If so then you can rescale by substitution of

    x = \frac{3}{4} y

    in the equation so it can be transformed to

    y^3 +2 y^2 +y -1 = 0.

    Sadly, if this is the case then there are no easy roots to find via easy analytical methods (unless you happen to know the cumbersome general solution for a cubic) so the only approach available would be to use a numerical method such as Newton-Raphson method.

    So the bottom line is that this will most likely need a numerical method.

    To do this I recommend that you first differentiate to find the maximum and minimum coordinates. From these values you should have a rough idea of where the single real root is. You can then put this first good guess into your Newton-Raphson iteration method to find the root.

    Finally, don't forget to work backwards to get back to s.
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