Simplify the expression The fraction is (x to the sixteenth power minus 1) over (x minus 1). by both factoring and using long division. On the answer, state which method you preferred, and why.

Answer 1

Factoring the expression:

The expression is (x^16 - 1) / (x - 1).

Now, we can factor the numerator using the difference of squares formula: a^2 - b^2 = (a + b)(a - b).

(x^16 - 1) = (x^8 + 1)(x^8 - 1) = (x^8 + 1)(x^4 + 1)(x^4 - 1) = (x^8 + 1)(x^4 + 1)(x^2 + 1)(x^2 - 1) = (x^8 + 1)(x^4 + 1)(x^2 + 1)(x + 1)(x - 1).

So, the simplified expression is (x^8 + 1)(x^4 + 1)(x^2 + 1)(x + 1).

Using long division:

For this method, we would need to divide x^16 - 1 by x - 1. This would involve multiple steps of dividing and subtracting, which can be more time-consuming compared to factoring.

I preferred the factoring method because it quickly broke down the expression into simpler factors, making it easier to simplify the expression. It also provides a clear understanding of how the expression was simplified.