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I know that we can determine the roots on the right-half plane by counting the sign changes in the first column. However, I'm unsure how to find the roots on the left-half plane. Is it the same as the right-half plane due to symmetry or am I missing something here?

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    \$\begingroup\$ You need to remember that Routh-Hurwitz tells how many roots in the right plane and in the left plane BUT doesn't tell you where are these roots. \$\endgroup\$ – CroCo Jul 23 at 19:04
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I think I figured it out. Total number of poles = order of characteristic equation so we can do:

Total number of poles - roots on the right-half plane (# of sign changes in first column) - roots on jw-axis = roots on the left-half plane

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