# To verify: The results provided below, 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 ) Also explain the factorization of A 5 − 1 . On a similar pattern obtain the factorization of A n − 1 , where n is a positive integer.

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1.3, Problem 136E
To determine

## To verify: The results provided below,  A2−1=(A−1)(A+1)A3−1=(A−1)(A2+A+1)A4−1=(A−1)(A3+A2+A+1)Also explain the factorization of A5−1 . On a similar pattern obtain the factorization of An−1 , where n is a positive integer.

Expert Solution

### Explanation of Solution

Given information:

The expressions,

A21=(A1)(A+1)A31=(A1)(A2+A+1)A41=(A1)(A3+A2+A+1)

Proof:

Consider the expressions,

A21=(A1)(A+1)A31=(A1)(A2+A+1)A41=(A1)(A3+A2+A+1)

Now, take one expression at a time and expand their right hand side.

For A21=(A1)(A+1) ,

(A1)(A+1)=A2+AA1=A21

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

(A1)(A2+A+1)=A3+A2+AA2A1=A31

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

(A1)(A3+A2+A+1)=A4+A3+A2+AA3A2A1=A41

Observe the pattern to factorize the polynomial A51 , first factor is of the form (A1) and second factor is a polynomial with degree one less than the polynomial.

A51=(A1)(A4+A3+A2+A+1)

Now, verify the expression obtained above,

(A1)(A4+A3+A2+A+1)=A5+A4+A3+A2+AA3A2A1=A51

Observe the pattern to factorize the polynomial An1 , first factor is of the form (A1) and second factor is a polynomial with degree one less than the polynomial.

An1=(A1)(An1+An1++A2+A+1)

Hence, the factorization of the polynomials of the form An1 has been verified.

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