Explain the Routh-Hurwitz stability criterion and how it is applied.

Routh-Hurwitz determines stability by examining the characteristic polynomial coefficients without solving for roots. A Routh array is constructed with first two rows from polynomial coefficients and subsequent rows calculated from determinants. The system is stable if all elements in the first column are positive. The number of sign changes equals the number of right-half-plane poles. Special cases (zero in first column, row of zeros) require special handling. It also determines the range of gain for stability.