Solution Found!
Use a proof by contradiction to show that there is no
Chapter 7, Problem 25E(choose chapter or problem)
Problem 25E
Use a proof by contradiction to show that there is no rational number r for which r3 + r+1 =0. [Hint: Assume that r = a/b is a root, where a and b are integers and is in lowest terms. Obtain an equation involving integers by multiplying by b3. Then look at whether a and b are each odd or even.]
Questions & Answers
QUESTION:
Problem 25E
Use a proof by contradiction to show that there is no rational number r for which r3 + r+1 =0. [Hint: Assume that r = a/b is a root, where a and b are integers and is in lowest terms. Obtain an equation involving integers by multiplying by b3. Then look at whether a and b are each odd or even.]
ANSWER:
Solution:
Step1
Given that
By using a proof by contradiction we have to show that there is no rational number r for which r3 + r+1 =0.
Step2
Suppose for use of contradiction that there is a rational number r where . Then by definition of rational numbers there are a and b such that
where a, b ∈ Z and a and b having no common factors.
Put value of we get