# Thread: Cubic Eqations

1. ## 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

2. The first polynomial obviously does not have positive roots because $\displaystyle s^3 + 6s^2 + 11s + 6 > 6$ 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 $\displaystyle f(s)=s^3 + s^2 - 8s - 12$, then $\displaystyle f(0)=-12<0$, but, say, $\displaystyle f(100) > 0$, so $\displaystyle f$ 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 $\displaystyle s^3 + s^2 - 8s - 12=(s-3)(s+2)^2$. One can use the rational root theorem to check if a polynomial has rational roots.