---
title: "R-H root distribution of s⁵+3s⁴+5s³+4s²+s+3"
subject: "Control Systems"
chapter: 4
key: "semester_3/control_systems:rh-s5-3-5-4-1"
source: https://www.bctengineer.com/q/semester_3/control_systems/rh-s5-3-5-4-1
site: BCT Engineering
---

# R-H root distribution of s⁵+3s⁴+5s³+4s²+s+3

Control Systems · Chapter 4: Stability

- Asked in: 2066 Bhadra (old course), 8 marks
- Video: [Example on Routh Array Stable System](https://www.youtube.com/watch?v=BApzhX2WBe0) — TutorialsPoint
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/rh-s5-3-5-4-1
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-4

## Question

Using R-H criteria, tell how many roots of polynomial are in right half plane, in left half plane and on jω-axis. P(s) = s⁵ + 3s⁴ + 5s³ + 4s² + s + 3.

## Short answer

Two roots lie in the right half-plane, three in the left half-plane, and none on the jω-axis. The system is unstable.

## Answer

**Result:** RHP = 2, LHP = 3, $j\omega$ axis = 0. The system is **unstable**.

Routh–Hurwitz criterion: the number of sign changes in the first column of the Routh array equals the number of roots in the right half plane. A complete zero row would indicate roots on the $j\omega$ axis.

$$
P(s) = s^5 + 3s^4 + 5s^3 + 4s^2 + s + 3
$$

#### Routh array

| Row | Col 1 | Col 2 | Col 3 |
|---|---|---|---|
| $s^5$ | 1 | 5 | 1 |
| $s^4$ | 3 | 4 | 3 |
| $s^3$ | $\frac{3(5)-1(4)}{3}=\frac{11}{3}$ | $\frac{3(1)-1(3)}{3}=0$ | |
| $s^2$ | $\frac{\frac{11}{3}(4)-3(0)}{\frac{11}{3}}=4$ | $\frac{\frac{11}{3}(3)-0}{\frac{11}{3}}=3$ | |
| $s^1$ | $\frac{4(0)-\frac{11}{3}(3)}{4}=-\frac{11}{4}$ | 0 | |
| $s^0$ | 3 | | |

First column: $1,\ 3,\ 3.67,\ 4,\ -2.75,\ 3$.

- $4 \to -2.75$: one sign change
- $-2.75 \to 3$: second sign change

#### Root distribution

No row became entirely zero, so no root lies on the $j\omega$ axis.

| Region | Number of roots |
|---|---|
| Right half plane | 2 |
| $j\omega$ axis | 0 |
| Left half plane | $5 - 2 = 3$ |

The roots ($\approx 0.256\pm j0.720$, $-0.941\pm j1.504$, $-1.631$) confirm this.

**Answer:** 2 roots in the RHP, 3 in the LHP, none on the $j\omega$ axis. The system is **unstable**.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For s^5+3s^4+5s^3+4s^2+s+3, how are the roots distributed?
   A. RHP 0, LHP 5, jω 0
   B. RHP 2, LHP 3, jω 0
   C. RHP 1, LHP 4, jω 0
   D. RHP 2, LHP 1, jω 2
   Answer: B. Two sign changes give 2 RHP roots; no zero row so none on jω; remaining 3 are in the LHP. (Key point: Root distribution)

2. In the s^3 row the second element is (3(1)-1(3))/3 = 0, but the first element is 11/3. Is this a problem?
   A. No, only the whole row being zero indicates symmetric roots; a zero in a later position is fine
   B. Yes, it means roots on the jω axis
   C. Yes, an epsilon must replace it
   D. Yes, the system is marginally stable
   Answer: A. Only the first-column zero or an entire zero row needs special treatment; here the row is not all zero and the first element is nonzero. (Key point: Zero in first column)

3. The first column is 1, 3, 3.67, 4, -2.75, 3. Where do the sign changes occur?
   A. 4 to -2.75 and -2.75 to 3
   B. 3 to 3.67 and 4 to -2.75
   C. Only 4 to -2.75
   D. None
   Answer: A. The signs go + + + + - +, giving two changes. (Key point: Sign changes)

4. What would a complete row of zeros in the array have indicated?
   A. All roots in the LHP
   B. A pole at infinity
   C. Perfect stability
   D. Roots symmetric about the origin, possibly on the jω axis
   Answer: D. An entire zero row signals symmetric roots, handled by the auxiliary polynomial; here none occurred so no jω roots. (Key point: Meaning of zero row)

## Close variants set in other exams

- [R-H root distribution of s⁶+2s⁵+8s⁴+12s³+20s²+16s+16](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arh-s6-2-8-12-20) — Control Systems, chapter 4, asked 4×
- [R-H root distribution of s⁵+4s⁴+2s³+8s²+s+4](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arh-s5-4-2-8-1) — Control Systems, chapter 4, asked 4×

## Related in Control Systems

- [R-H root distribution of s⁶+3s⁵+4s⁴+6s³+5s²+3s+2](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arh-s6-3-4-6-5) — Control Systems, chapter 4, asked 5×
- [K and T for roots left of s=−a](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arh-k-t-left-of-a) — Control Systems, chapter 4
- [R-H root distribution for 200/[s(s³+6s²+11s+6)]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arh-200-s-s3-6-11) — Control Systems, chapter 4
- [Bode plot of 48(1+s)/[s²(1+3s)(1+0.5s)(2+0.2s)]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abode-48-1s-s2-1-3s) — Control Systems, chapter 6
- [State space advantages; state model of (2s²+3s+1)/(s³+5s²+6s+7)](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-adv-tf-2s2-3s-1) — Control Systems, chapter 8
- [Root locus of K(s+2)/(s²−4s+13)](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-k-s2-s2-4s-13) — Control Systems, chapter 5

## Related in other subjects

- [Bode plot: 20(S+1)/((S²+4S+2)(S²+5S))](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Abode-20s1-s2-4s-2-s2) — Electrical Circuit Theory, chapter 5, asked 4×
- [Parabolic arch 100 m: UDL left half and 80 kN](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aarch-100m-80kn-udl-d-20m) — Theory of Structures I, chapter 6, asked 4×