Proof Assume that 2 | (x² − 1). So x² − 1 = 2y for some integer y. Thus, x² = 2y + 1 is an odd integer. It then follows by Theorem 3.12 that x too is odd. Hence, x = 2z + 1 for some integer z. Then x² − 1 = (2z + 1)² − 1 = (4z² + 4z + 1) − 1 = 4z² + 4z = 4(z² + z). Since z² + z is an integer, 4 | (x² − 1).
Proof Assume that 2 | (x² − 1). So x² − 1 = 2y for some integer y. Thus, x² = 2y + 1 is an odd integer. It then follows by Theorem 3.12 that x too is odd. Hence, x = 2z + 1 for some integer z. Then x² − 1 = (2z + 1)² − 1 = (4z² + 4z + 1) − 1 = 4z² + 4z = 4(z² + z). Since z² + z is an integer, 4 | (x² − 1).
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.2: Properties Of Division
Problem 51E
Related questions
Question
Please if able explain the following example in detail, I dont get the substitution and am very confused.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 5 steps with 4 images
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning