Simplify the following expressions in the indicated polynomial rings: A. (3x³ + 2x² + 4x + 2) + (2x¹ + 2x³ + x + 1) in Z5 [x] B. (x + 1)³ in Z3 [x] C. (x − 1)5 in Z5 [x] D. (x²-3x + 2)(2x³ - 4x + 1) in Z7 [x]
Simplify the following expressions in the indicated polynomial rings: A. (3x³ + 2x² + 4x + 2) + (2x¹ + 2x³ + x + 1) in Z5 [x] B. (x + 1)³ in Z3 [x] C. (x − 1)5 in Z5 [x] D. (x²-3x + 2)(2x³ - 4x + 1) in Z7 [x]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem: Simplify the following expressions in the indicated polynomial rings:**
A. \( (3x^3 + 2x^2 + 4x + 2) + (2x^4 + 2x^3 + x + 1) \) in \(\mathbb{Z}_5[x]\)
B. \( (x + 1)^3 \) in \(\mathbb{Z}_3[x]\)
C. \( (x - 1)^5 \) in \(\mathbb{Z}_5[x]\)
D. \( (x^2 - 3x + 2)(2x^3 - 4x + 1) \) in \(\mathbb{Z}_7[x]\)
**Explanation:**
- Polynomials are simplified in specific modulus rings, denoted by \(\mathbb{Z}_n[x]\), where all coefficients are reduced modulo \(n\).
- For example, in \(\mathbb{Z}_5[x]\), each coefficient of the polynomial is calculated modulo 5.
- These operations ensure that all resulting polynomial coefficients are within the set \(\{0, 1, \ldots, n-1\}\).
To solve each part:
1. **Part A:** Combine like terms and reduce each coefficient modulo 5.
2. **Part B:** Expand the power and reduce coefficients modulo 3.
3. **Part C:** Similar to Part B, but reduce modulo 5.
4. **Part D:** Multiply the polynomials and reduce each coefficient modulo 7.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feabade52-5174-4cc3-8205-365538407c90%2Faf0890c1-5320-4502-89a2-1d4312792312%2Fdyftcce_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem: Simplify the following expressions in the indicated polynomial rings:**
A. \( (3x^3 + 2x^2 + 4x + 2) + (2x^4 + 2x^3 + x + 1) \) in \(\mathbb{Z}_5[x]\)
B. \( (x + 1)^3 \) in \(\mathbb{Z}_3[x]\)
C. \( (x - 1)^5 \) in \(\mathbb{Z}_5[x]\)
D. \( (x^2 - 3x + 2)(2x^3 - 4x + 1) \) in \(\mathbb{Z}_7[x]\)
**Explanation:**
- Polynomials are simplified in specific modulus rings, denoted by \(\mathbb{Z}_n[x]\), where all coefficients are reduced modulo \(n\).
- For example, in \(\mathbb{Z}_5[x]\), each coefficient of the polynomial is calculated modulo 5.
- These operations ensure that all resulting polynomial coefficients are within the set \(\{0, 1, \ldots, n-1\}\).
To solve each part:
1. **Part A:** Combine like terms and reduce each coefficient modulo 5.
2. **Part B:** Expand the power and reduce coefficients modulo 3.
3. **Part C:** Similar to Part B, but reduce modulo 5.
4. **Part D:** Multiply the polynomials and reduce each coefficient modulo 7.
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