▁conditions ▁are ▁satisfied : ▁▁ 1 . ▁P ( s ) ▁is ▁real ▁when ▁s ▁is ▁real . ▁▁▁ 2 . ▁The ▁roots ▁of ▁P ( s ) ▁have ▁real ▁parts ▁which ▁are ▁zero ▁or ▁negative . ▁ ▁Hur witz ▁polynomials ▁are ▁important ▁in ▁control ▁systems ▁theory , ▁because ▁they ▁represent ▁the ▁characteristic ▁equations ▁of ▁stable ▁linear ▁systems . ▁ ▁Whether ▁a ▁polynomial ▁is ▁Hur witz ▁can ▁be ▁determined ▁by ▁solving ▁the ▁equation ▁to ▁find ▁the ▁roots , ▁or ▁from ▁the ▁coefficients ▁without ▁solving ▁the ▁equation ▁by ▁the ▁R outh – H ur witz ▁stability ▁criter ion . ▁ ▁Ex amples ▁▁ ▁A ▁simple ▁example ▁of ▁a ▁Hur witz ▁polynomial ▁is ▁the ▁following : ▁ ▁The ▁only ▁real ▁solution ▁is ▁− 1 , ▁as ▁it ▁factors ▁to ▁ ▁In ▁general , ▁all ▁second - deg ree ▁polynomials ▁with ▁positive ▁coefficients ▁are ▁Hur witz . ▁This ▁follows ▁directly ▁from ▁the ▁quadratic ▁formula : ▁ ▁where , ▁if ▁the ▁determin ant ▁b ^ 2 - 4 ac ▁is ▁less ▁than ▁zero , ▁then ▁the ▁polynomial ▁will ▁have ▁two ▁complex - con j ug ate ▁solutions ▁with ▁real ▁part ▁- b / 2 a , ▁which ▁is ▁negative ▁for ▁positive ▁a ▁and ▁b . ▁If ▁it ▁is ▁equal ▁to ▁zero , ▁there ▁will ▁be ▁two ▁coinc iding ▁real ▁solutions ▁at ▁- b / 2 a . ▁Finally , ▁if ▁the ▁determin ant ▁is ▁greater ▁than ▁zero , ▁there ▁will ▁be ▁two ▁real ▁negative ▁solutions , ▁because ▁ ▁for ▁positive ▁a , ▁b ▁and ▁c . ▁