Use a proof by contradiction to show that there is no

Chapter 7, Problem 25E

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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.]

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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

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