Routh Hurwitz criterion with wrong estimation of the number of RHP poles with epsilon approach

Question

I'm studying Routh-Hurwitz criterion and I found that for some specific cases the criterion estimates the wrong number of RHP (right half plane) poles when using epsilon approach. Some cases where the order is equal or higher then 5 and the second highest s power is missing, leads to a situation where the number of sign changes doesn't correspond to the number of RHP poles. For example,

$$7s^5 + 6s^3 + 7s^2 + 3s + 53 = 0$$

would give (with epsilon approach):

Routh's table

Four sign changes = 4 RHP poles, but this polynom only has 2 RHP poles (Wolfram Alpha solution of this equation)! Am I missing something or is it just limitation of epsilon approach?

Answer 1

The correct Routh table is shown below. Whether you assume the sign of $\epsilon$ positive or negative, you will end up with two sign changes. Try it.