Problem:
Find the remainder when x^45 is divided by (x^2 - 1).

Solution:
To find the remainder when x^45 is divided by (x^2 - 1), we can use the polynomial division method.

Step 1:
Write the dividend and divisor in descending powers of x:
Dividend: x^45
Divisor: x^2 - 1

Step 2:
Divide the first term of the dividend (x^45) by the first term of the divisor (x^2) to get the quotient term:
Quotient term: x^45 / x^2 = x^43

Step 3:
Multiply the divisor (x^2 - 1) by the quotient term (x^43) and subtract it from the dividend:
x^45 - x^43(x^2 - 1)

Step 4:
Simplify the expression obtained in step 3:
x^45 - x^45 + x^43
= x^43

Step 5:
Repeat steps 2-4 until the degree of the remaining expression is less than the degree of the divisor.

In this case, the degree of the remaining expression (x^43) is still greater than the degree of the divisor (x^2 - 1), so we need to continue.

Step 6:
Divide the first term of the remaining expression (x^43) by the first term of the divisor (x^2) to get the next quotient term:
Quotient term: x^43 / x^2 = x^41

Step 7:
Multiply the divisor (x^2 - 1) by the quotient term (x^41) and subtract it from the remaining expression:
x^43 - x^41(x^2 - 1)

Step 8:
Simplify the expression obtained in step 7:
x^43 - x^43 + x^41
= x^41

Step 9:
Repeat steps 6-8 until the degree of the remaining expression is less than the degree of the divisor.

Continue this process until the degree of the remaining expression is less than the degree of the divisor.

Eventually, we will reach a point where the degree of the remaining expression is less than the degree of the divisor. The remainder at this point will be the final remainder when x^45 is divided by (x^2 - 1).

Conclusion:
After performing the polynomial division process, we find that the remainder when x^45 is divided by (x^2 - 1) is x^41.

Note:
The above explanation is a step-by-step approach to finding the remainder when x^45 is divided by (x^2 - 1). The process can be generalized for any division of polynomials.