find the remainder when x^99 - 99 is divided by x^2 -1

Answer 1

To find the remainder when \(x^{99} - 99\) is divided by \(x^2 - 1\), we first need to factor the denominator.

\(x^2 - 1\) can be factored as \((x + 1)(x - 1)\).

Now, we can rewrite \(x^{99} - 99\) as \(x^{99} - x + x - 99\).

Using the factorization of the denominator, we rewrite the expression as follows:

\(x^{99} - x + x - 99 = x^{99} - x + x - 99(x^2 - 1) + 99(x^2 - 1)\)

Expanding, we get:

\(x^{99} - x + x - 99(x^2 - 1) + 99(x^2 - 1) = x^{99} - 99x - 99 + 99x + 99 = x^{99} - 99\)

Therefore, the remainder when \(x^{99} - 99\) is divided by \(x^2 - 1\) is \(x^{99} - 99\).