Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 7: Problem 517

If \(a, b \in N, a \neq b\) then for \(r \in N, a^{n}-b^{n}\) is divisible by (a) \(a-b\) (b) \(\mathrm{b}-\mathrm{a}\) (c) both (a) and (b) (d) None of these

Short Answer

Expert verified
Using the factorization of the difference of two powers, it can be shown that \(a^n - b^n\) is divisible by both \(a - b\) and \(b - a\). Therefore, the correct answer is (c) both (a) and (b).

Step by step solution

01

Write the factorization of the difference of two powers

The difference of two powers can be factored as follows: \[a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + a^{n-3}b^2 + \cdots + b^{n-1})\]
02

Analyze the divisibility by \(a - b\)

As shown in Step 1, \(a - b\) is already a factor of \(a^n - b^n\), since the entire expression can be factored into two terms, one of which is \((a - b)\). Therefore, \(a^n - b^n\) is divisible by \(a - b\).
03

Analyze the divisibility by \(b - a\)

Notice that we can rewrite \(b - a\) as \(-(a - b)\). Thus, if we factor \(-1\) out of the expression in Step 1, we have: \[a^n - b^n = -(b - a)(-a^{n-1} - a^{n-2}b - a^{n-3}b^2 - \cdots - b^{n-1})\] Now, it is clear that \(b - a\) is also a factor of \(a^n - b^n\). Therefore, \(a^n - b^n\) is divisible by \(b - a\) as well.
04

Conclusion

Since \(a^n - b^n\) is divisible by both \(a - b\) and \(b - a\), the correct answer is (c) both (a) and (b).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The Co-efficient of \(x^{3}\) in \(\left(1-x+x^{2}\right)^{5}\) is (a) \(-30\) (b) \(-20\) (c) \(-10\) (d) 30

Coefficients of \((2 r+4)\) th term and \((r-2)\) th term are equal in expansion of \((1+x)^{18}\) then \(r=\) (a) 4 (b) 5 (c) 6 (d) 7

If \(\mathrm{P}\) and \(\mathrm{Q}\) are coefficients of \(\mathrm{x}^{\mathrm{n}}\) in expansion of \((1+\mathrm{x})^{2 \mathrm{n}}\) and \((1+x)^{2 n-1}\) then (a) \(\mathrm{P}=\mathrm{Q}\) (b) \(P=2 Q\) (c) \(2 \mathrm{P}=\mathrm{Q}\) (a) \(P+Q=0\)

The constant term in expansion of \(\left[\left\\{(x+1) /\left\\{x^{(2 / 3)}-x^{(1 / 3)}+1\right\\}\right\\}-\left\\{(x-1) /\left(x-x^{(1 / 2)}\right)\right\\}\right]^{10}\) is (a) 210 (b) 105 (c) 70 (d) 35

\(3^{\mathrm{rd}}\) term in expansion of \(\left[(1 / \mathrm{x})+\mathrm{x}^{(\log )}{ }_{10}(\mathrm{x})\right]^{5}\) is 1000 then \(\mathrm{x}=\) (a) 10 (b) 100 (c) 1000 (d) None
See all solutions

Recommended explanations on English Textbooks

Language Analysis

Read Explanation

Argumentative Essay

Read Explanation

Free Response Essay

Read Explanation

Graphology

Read Explanation

Essay Prompts

Read Explanation

5 Paragraph Essay

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free