For a continuous-time closed-loop system, the Routh-Hurwitz criterion counts right-half-plane roots of the characteristic polynomial without solving explicitly for every root.

For

$$a_3s^3+a_2s^2+a_1s+a_0,$$

the Routh array begins

$$\begin{array}{c|cc} s^3&a_3&a_1\ s^2&a_2&a_0\ s^1&\dfrac{a_2a_1-a_3a_0}{a_2}&0\ s^0&a_0& \end{array}$$

Higher-order arrays are generated by the same elimination pattern from the two rows above.

Under the standard nondegenerate construction, the number of sign changes down the first column equals the number of roots in the open right half-plane.

A stable polynomial therefore has no first-column sign changes and no roots on the imaginary axis. For the cubic above, assuming $a_3>0$, stability requires the first-column entries to keep the same positive sign.

When controller gain appears symbolically in the coefficients, these sign conditions can give the entire range of gains that preserves stability.