Consider the quotient ring Q[x]/< x² + x – 2 > (a) Give an example of a zero divisor in this ring. Give the multiplication which shows it is a zero divisor. (b) Prove that if a + bx + < x² + x- 2 >= c + dx + < x? +x – 2 >, then a = c and b = d (c) Determine (x + < x² + x – 2 >)-1 as follows: (i) Briefly explain why (x + < x² + x – 2 >)-1 = a + bx + < x² + x – 2 >, for some | a, b E Q (ii) Use part (b) to determine a and b (iii) Verify that your answer is in fact the inverse.
Consider the quotient ring Q[x]/< x² + x – 2 > (a) Give an example of a zero divisor in this ring. Give the multiplication which shows it is a zero divisor. (b) Prove that if a + bx + < x² + x- 2 >= c + dx + < x? +x – 2 >, then a = c and b = d (c) Determine (x + < x² + x – 2 >)-1 as follows: (i) Briefly explain why (x + < x² + x – 2 >)-1 = a + bx + < x² + x – 2 >, for some | a, b E Q (ii) Use part (b) to determine a and b (iii) Verify that your answer is in fact the inverse.
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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![Consider the quotient ring Q[x]/< x² + x – 2 >
(a) Give an example of a zero divisor in this ring. Give the multiplication which shows it is a
zero divisor.
(b) Prove that if a + bx + < x² + x – 2 >= c + dx + < x² + x – 2 >, then a = c and b = d
(c) Determine (x + < x² + x – 2 >)-1 as follows:
(i) Briefly explain why (x + < x? + x – 2 >)-1 = a + bx + < x² + x – 2 >, for some
a, b E Q
(ii) Use part (b) to determine a and b
(iii) Verify that your answer is in fact the inverse.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe02001c2-7a8a-499c-b7d7-28eaa52a5370%2Fdd7c23be-6dce-455c-9dae-9ebaddb7460b%2Fdldxblb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the quotient ring Q[x]/< x² + x – 2 >
(a) Give an example of a zero divisor in this ring. Give the multiplication which shows it is a
zero divisor.
(b) Prove that if a + bx + < x² + x – 2 >= c + dx + < x² + x – 2 >, then a = c and b = d
(c) Determine (x + < x² + x – 2 >)-1 as follows:
(i) Briefly explain why (x + < x? + x – 2 >)-1 = a + bx + < x² + x – 2 >, for some
a, b E Q
(ii) Use part (b) to determine a and b
(iii) Verify that your answer is in fact the inverse.
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