91Ó°ÊÓ

We can see that both terms in the numerator, \(x^{2n}\) and 1, are squares. Similarly, in the denominator, \(x^2\) and 1 are squares. So, both the numerator and the denominator are in the difference of squares form.

Recall the difference of squares formula: \(a^2 - b^2 = (a + b)(a - b)\). Apply this formula to both the numerator and the denominator of the given function: Numerator: \(x^{2n} - 1 = (x^n + 1)(x^n - 1)\) Denominator: \(x^2 - 1 = (x + 1)(x - 1)\) Now, rewrite the function using the factored expressions: $$f(x) = \frac{x^2\left((x^n + 1)(x^n - 1)\right)}{(x + 1)(x - 1)}$$

At this point, we need to check if there are any common factors between the numerator and the denominator that can be cancelled out. Notice that the \((x^n - 1)\) term in the numerator is a difference of squares itself, so we can factor it again: $$x^n - 1 = (x^{\frac{n}{2}} + 1)(x^{\frac{n}{2}} - 1)$$ Now, rewrite the function using the factored expressions: $$f(x) = \frac{x^2\left((x^n + 1)(x^{\frac{n}{2}} + 1)(x^{\frac{n}{2}} - 1)\right)}{(x + 1)(x - 1)}$$ In this simplified form, there are no common factors between the numerator and the denominator that can be cancelled out. Therefore, the function is now in its simplest form: $$f(x) = \frac{x^2\left((x^n + 1)(x^{\frac{n}{2}} + 1)(x^{\frac{n}{2}} - 1)\right)}{(x + 1)(x - 1)}$$