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DISCOVER: Factoring A n − 1 (a) Verify the following formulas by expanding and simplifying he right-hand side. A 2 − 1 = ( A − 1 ) ( A + 1 ) A 3 − 1 = ( A − 1 ) ( A 2 + A + 1 ) A 4 − 1 = ( A − 1 ) ( A 3 + A 2 + A + 1 ) (b) On the basis of the pattern displayed in this list, how do you think A 5 − 1 would factor? Verify your conjucture. Now generalize the pattern you have observed to obtain a factoring formula for A n − 1 , where n is a positive integer.

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Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
Publisher: Cengage Learning
ISBN: 9781305071742

Solutions

Chapter
Section
BuyFindarrow_forward

Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
Publisher: Cengage Learning
ISBN: 9781305071742
Chapter P.6, Problem 120E
Textbook Problem
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DISCOVER: Factoring A n 1

(a) Verify the following formulas by expanding and simplifying he right-hand side.

A 2 1 = ( A 1 ) ( A + 1 ) A 3 1 = ( A 1 ) ( A 2 + A + 1 ) A 4 1 = ( A 1 ) ( A 3 + A 2 + A + 1 )

(b) On the basis of the pattern displayed in this list, how do you think A 5 1 would factor? Verify your conjucture. Now generalize the pattern you have observed to obtain a factoring formula for A n 1 , where n is a positive integer.

To determine

(a)

To find:

To verify the following formulas by expanding and simplifying he right-hand side.

Explanation of Solution

Consider the following expression,

A21=(A1)(A+1)

First multiply the terms in RHS to get following,

(A1)(A+1)=A2A+A12=A2+(AA)1=A21

Consider the following expression,

A31=(A1)(A2+A+1)

First multiply the terms in RHS to get following,

(A1)(A2+A+1)=A3+A2+AA2A12=A3+(A2A2)+(AA)12=A31

Consider the following expression,

A41=(A1)(A3+A2+A+1)

First multiply the terms in RHS to get following,

To determine

(b)

To find:

To verify the following,

A51 and to obtain a factoring formula for An1, where n is a positive integer.

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