---
title: "Find x(2), x(3) from X(z)=(2z²+3z+12)/(z-1)⁴"
subject: "Applied Mathematics"
chapter: 2
key: "semester_4/applied_mathematics:inverse-z-find-x2-x3"
source: https://www.bctengineer.com/q/semester_4/applied_mathematics/inverse-z-find-x2-x3
site: BCT Engineering
---

# Find x(2), x(3) from X(z)=(2z²+3z+12)/(z-1)⁴

Applied Mathematics · Chapter 2: The Z-Transform

- Asked in: 2071 Magh, 5 marks
- Video: [Z-Transform Problem on Initial Value Theorem | Find u₂ and u₃ | BMATEC301 | Mathematics-III | ](https://www.youtube.com/watch?v=v8a7yf6_bCA) — 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-find-x2-x3
- Whole chapter: https://www.bctengineer.com/md/semester_4/applied_mathematics/chapter-2

## Question

If Z[x(k)] = (2z² + 3z + 12)/(z - 1)⁴, find the value of x(2) and x(3).

## Short answer

Expand X(z) in powers of z^{-1}: X(z)=(2z^{-2}+3z^{-3}+12z^{-4})/(1-z^{-1})^4. The coefficients give x(2)=2 and x(3)=11.

## Answer

**Idea.** By definition X(z) = x(0) + x(1)z⁻¹ + x(2)z⁻² + x(3)z⁻³ + … So x(k) is the coefficient of z⁻ᵏ in the expansion of X(z) in powers of 1/z (long division / binomial series).

**Step 1: write in powers of w = z⁻¹.**

```text
X(z) = (2z² + 3z + 12) / (z − 1)⁴
     = (2z² + 3z + 12) / z⁴  ·  1/(1 − w)⁴
     = (2w² + 3w³ + 12w⁴) · 1/(1 − w)⁴
```

**Step 2: binomial series.**

1/(1 − w)⁴ = 1 + 4w + 10w² + 20w³ + 35w⁴ + …

**Step 3: multiply.**

```text
coefficient of w⁰ : 0                        → x(0) = 0
coefficient of w¹ : 0                        → x(1) = 0
coefficient of w² : 2                        → x(2) = 2
coefficient of w³ : 2·4 + 3  = 11            → x(3) = 11
coefficient of w⁴ : 2·10 + 3·4 + 12 = 44     → x(4) = 44
```

**Answer:** x(2) = 2 and x(3) = 11.

## Check yourself

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

1. x(k) is
   A. the coefficient of z^-k in the expansion of X(z) in powers of 1/z
   B. the coefficient of z^k
   C. the residue at z = 1
   D. X(k)
   Answer: A. X = sum x(k) z^-k. (Key point: Idea)

2. (2z^2+3z+12)/(z-1)^4 in w = 1/z is
   A. (2w^2 + 3w + 12)/(1-w)^4
   B. (2w^2 + 3w^3 + 12w^4)/(1-w)^4
   C. (2 + 3w + 12w^2)/(1-w)^4
   D. (2w + 3w^2 + 12w^3)/(1-w)^4
   Answer: B. Divide by z^4. (Key point: Rewrite)

3. 1/(1-w)^4 =
   A. 1 + w + w^2 + ...
   B. 1 + 4w + 6w^2 + ...
   C. 1 + 4w + 10w^2 + 20w^3 + ...
   D. 1 + 2w + 3w^2 + ...
   Answer: C. Coefficients are C(k+3,3). (Key point: Binomial series)

4. x(2) and x(3) are
   A. 2 and 8
   B. 3 and 11
   C. 0 and 2
   D. 2 and 11
   Answer: D. x(2) = 2; x(3) = 2*4 + 3 = 11. (Key point: Result)

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