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Math Help - Cubic Eqations

  1. #1
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    Cubic Eqations

    Systems are unstable if their characteristic equations have a positive root (solution). Determine whether each of the following characteristic equations represents a stable or unstable system.

    (i) s3 + 6s2 + 11s + 6 = 0


    (ii) s3 + s2 - 8s - 12 = 0
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  2. #2
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    The first polynomial obviously does not have positive roots because s^3 + 6s^2 + 11s + 6 > 6 when s > 0.

    For the second polynomial, there are several ways to reason. First, Descartes' rule of signs says that it has exactly one positive root. One can also note that if f(s)=s^3 + s^2 - 8s - 12, then f(0)=-12<0, but, say, f(100) > 0, so f must have an x-intercept. I am sure there are other ways; what you want probably depends on what methods you have studied.

    Edit: With a bit of trial-and error (or some factorization tools) one can see that s^3 + s^2 - 8s - 12=(s-3)(s+2)^2. One can use the rational root theorem to check if a polynomial has rational roots.
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