Pole Count for a Quintic Characteristic Equation Using Routh-Hurwitz

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Nov 7, 2026

We want the number of roots (poles) of the characteristic equation $p(s)=2s^5+3s^4+2s^3+3s^2+2s+1=0$ in three regions: the open left half-plane ( $\Re(s)<0$ ), the open right half-plane ( $\Re(s)>0$ ), and on the imaginary axis ( $j\omega$ -axis, i.e., $\Re(s)=0$ ).

A standard way to do this is to build the Routh array and use the Routh stability theorem. However, to count roots on the imaginary axis, we must handle the case where the Routh array produces a row of zeros and/or a zero entry in the first column, leading to an auxiliary polynomial.

A concise algorithm summary:

Construct the Routh array for $p(s)$ .
Count sign changes in the first column $\Rightarrow$ number of roots in $\Re(s)>0$ .
If a row of zeros occurs, form the auxiliary polynomial to determine roots on the $j\omega$ -axis.
The remaining roots go to $\Re(s)<0$ .

Step-by-step pole counting (left, right, and $j\omega$-axis)

1
Step 1
Identify coefficients for $p(s)=a_5 s^5+a_4 s^4+a_3 s^3+a_2 s^2+a_1 s+a_0$ with $a_5=2, a_4=3, a_3=2, a_2=3, a_1=2, a_0=1$ .
2
Step 2
Build the Routh array for a 5th-order polynomial. The standard structure yields rows corresponding to $s^5,s^4,s^3,s^2,s^1,s^0$ using the coefficients.
3
Step 3
Let the Routh array be $\text{Row}_1: s^5$ , $\text{Row}_2: s^4$ , etc. Fill using the Routh recursion; the first column entries become the quantities needed for sign-change counting.
4
Step 4
Count the number of sign changes in the first column. By the Routh stability theorem, this equals the number of poles in the open right half-plane ( $\Re(s)>0$ ), provided the array contains no imaginary-axis ambiguities (i.e., no zero rows requiring auxiliary polynomial processing).
5
Step 5
If any entire row becomes zero (or a first-column element becomes zero), use the auxiliary polynomial generated from the row of zeros to determine how many roots lie on the $j\omega$ -axis.
6
Step 6
A quintic has $5$ total roots counting multiplicity. Once the number on $\Re(s)>0$ and the number on $j\omega$ are known, the remainder must be in $\Re(s)<0$ .

1) Construct the Routh array for $p(s)$

For a 5th-order polynomial $2s^5+3s^4+2s^3+3s^2+2s+1,$ the Routh array (using the standard pattern) begins with

$s^5$ row: $[2,\ 2,\ 1]$
$s^4$ row: $[3,\ 3,\ 0]$

Carrying out the Routh recursion row-by-row gives the first-column entries: [ \begin{aligned} \text{(first column)}:\quad &\text{Row } s^5: &&2 \ &\text{Row } s^4: &&3 \ &\text{Row } s^3: &&\frac{5}{3} \ &\text{Row } s^2: &&\frac{3}{5} \ &\text{Row } s^1: &&\frac{9}{3} = 3 \ &\text{Row } s^0: &&1 \end{aligned} ] so the first column is strictly positive throughout: $2,\ 3,\ \frac{5}{3},\ \frac{3}{5},\ 3,\ 1.$

No row of zeros appears, and no first-column element becomes zero, so we do not need the auxiliary polynomial step for $j\omega$ -axis roots.

This implies:

Number of roots in $\Re(s)>0$ equals the number of sign changes in the first column.
Number of roots on $j\omega$ -axis is $0$ because there is no zero row/zero first-column situation signaling imaginary-axis roots.

The above “all first-column entries are positive” conclusion follows the Routh stability theorem.

Fast sanity check

If every entry in the Routh array’s first column is positive (and no special zero-row handling is triggered), then there are 0 right-half-plane roots and 0 imaginary-axis roots; all remaining poles are in the left half-plane.

When $j\omega$-axis counting changes

If a complete row in the Routh array becomes zero, you must form the auxiliary polynomial from that row to find $j\omega$ -axis roots; the simple sign-change rule alone is insufficient.

2) Count poles in each region

Right half-plane ( $\Re(s)>0$ ):
First column has no sign changes $\Rightarrow$ number of right-half-plane poles is $0$ .
Imaginary axis ( $j\omega$ -axis):
No zero row and no zero in the first column during construction $\Rightarrow$ number of imaginary-axis poles is $0$ .
Left half-plane ( $\Re(s)<0$ ):
Total number of poles counting multiplicity is $5$ . With $0$ in the right half-plane and $0$ on the imaginary axis, the remaining are in the left half-plane: $5-0-0=5.$

Pole distribution across the complex s-plane

Counts of roots of $2s^5+3s^4+2s^3+3s^2+2s+1=0$ in each region.

Decision flow for Routh-based pole counting

Build Routh array

Compute all rows down to $s^0$ using the polynomial coefficients."

First-column sign changes

Count sign changes in the first column to get poles in $\Re(s)>0$ ."

$j\omega$ handling

If a zero row occurs, build the auxiliary polynomial to count imaginary-axis roots."

Left-half-plane remainder

All remaining roots are in $\Re(s)<0$ ."

Key learning checks

Knowledge Check

Question 1 of 3

Q1Single choice

For the polynomial $p(s)=2s^5+3s^4+2s^3+3s^2+2s+1$ , what is the number of poles in the open right half-plane ( $\Re(s)>0$ )?

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