If given a circuit composed only of resistors, inductors, and capacitors, if its impulse response decays over time, there should be negative exponents in that response. But that means that the complex roots of the characteristic polynomial should have negative (or zero) real parts. I assumed this was because the characteristic polynomial has positive coefficients but I later found out that not all polynomials with positive coefficients have no positive real parts in their roots. For instance, take a look at $$\s^5+s^4+s^3+s^2+s+1\$$.