---
title: "Inverse z of z/(z²-3z+2)"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:inverse-z-z-over-z2-3z-2"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/inverse-z-z-over-z2-3z-2
site: BCT Engineering
---

# Inverse z of z/(z²-3z+2)

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2070 Bhadra, 5 marks; 2069 Bhadra (asked 2 times)
- Video: [Inverse Z-Transform of z/((z−2)(z−3)) | VTU Exam](https://www.youtube.com/watch?v=Bta7Bd_uIxQ) — Mathematics Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/applied_mathematics/inverse-z-z-over-z2-3z-2
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

Find the inverse z-transform of z/(z² - 3z + 2).

## Short answer

X(z)/z partial fractions yield X(z)=z/(z−2)−z/(z−1). Inverting: x(k)=2^k−1, k≥0.

## Answer

X(z) = z/(z² − 3z + 2) = z/((z − 1)(z − 2)).

**Step 1.** Partial fractions on X(z)/z:

X(z)/z = 1/((z − 1)(z − 2)) = A/(z − 1) + B/(z − 2)

- A = 1/(z − 2) at z = 1 = −1
- B = 1/(z − 1) at z = 2 = 1

**Step 2.**

X(z) = z/(z − 2) − z/(z − 1)

**Step 3.** Using Z⁻¹[z/(z − a)] = aᵏ:

**Answer: x(k) = 2ᵏ − 1,  k ≥ 0**

```text
k    : 0  1  2  3   4
x(k) : 0  1  3  7  15
```

Check by long division: X(z) = z⁻¹ + 3z⁻² + 7z⁻³ + 15z⁻⁴ + … ✓

## Check yourself

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

1. Factor z²−3z+2.
   A. (z+1)(z+2)
   B. (z−1)(z−2)
   C. (z−1)(z+2)
   D. (z−3)(z+2)
   Answer: B. Roots 1 and 2. (Key point: Factorisation)

2. X(z) in partial fractions?
   A. z/(z−1) − z/(z−2)
   B. z/(z−2) + z/(z−1)
   C. z/(z−2) − z/(z−1)
   D. 1/(z−2) − 1/(z−1)
   Answer: C. A = −1, B = 1. (Key point: Partial fractions)

3. x(k)?
   A. 1 − 2^k
   B. 2^k + 1
   C. 2^{k−1}
   D. 2^k − 1
   Answer: D. x = 0, 1, 3, 7, ... (Key point: Result)

## Related in Applied Mathematics

- [Inverse z of (3z³+2z)/((z-3)²(z-2)), inversion integral](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Ainverse-z-3z3-2z-inversion) — Applied Mathematics, chapter 2, asked 3×
- [Inverse z of z(3z²-6z+4)/((z-1)²(z-2))](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Ainverse-z-3z2-6z-4) — Applied Mathematics, chapter 2
- [Inverse z of (2z³+z)/((z-2)²(z-1))](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Ainverse-z-2z3-z) — Applied Mathematics, chapter 2, asked 5×
- [Inverse z of (z-4)/((z-1)(z-2)²)](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Ainverse-z-z-4-over-z-1-z-2-sq) — Applied Mathematics, chapter 2, asked 5×
- [Inverse z of (2z²-5z)/((z-2)(z-3))](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Ainverse-z-2z2-5z) — Applied Mathematics, chapter 2, asked 5×
- [Inverse z of (z-4)/((z-1)(z-2)²), inversion integral](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Ainverse-z-z-4-inversion-integral) — Applied Mathematics, chapter 2, asked 3×

## Related in other subjects

- [Inverse z-transform, ROC |z| < 0.4](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Ainverse-z-roc-less-0-4) — Digital Signal Analysis and Processing, chapter 2, asked 5×
- [Inverse Z-transform by inversion integral](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Ainverse-integral-step-exponential) — Digital Control System, chapter 2, asked 10×
- [Inverse Z-transform of z(z+2)/(z−1)²](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Ainverse-integral-z-z-plus-2) — Digital Control System, chapter 2, asked 10×
- [Inverse Z-transform: double pole at z=1](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Ainverse-integral-z2-double-pole) — Digital Control System, chapter 2, asked 10×