1. **State the problem:** Express the rational function $$\frac{x^4 - 1}{x^2 - 1}$$ in partial fractions. 2. **Recall the formula and rules:** Partial fraction decomposition is used to express a rational function as a sum of simpler fractions, typically when the degree of the numerator is less than the degree of the denominator. If the numerator's degree is greater or equal, perform polynomial division first. 3. **Perform polynomial division:** Since the numerator degree (4) is greater than the denominator degree (2), divide $$x^4 - 1$$ by $$x^2 - 1$$. $$x^4 - 1 \div x^2 - 1$$ Divide the leading terms: $$\frac{x^4}{x^2} = x^2$$ Multiply back: $$x^2(x^2 - 1) = x^4 - x^2$$ Subtract: $$\begin{aligned} (x^4 - 1) - (x^4 - x^2) &= x^4 - 1 - x^4 + x^2 \\ &= x^2 - 1 \end{aligned}$$ 4. **Repeat division with remainder:** Now divide the remainder $$x^2 - 1$$ by $$x^2 - 1$$. $$\frac{x^2 - 1}{x^2 - 1} = 1$$ Multiply back: $$1 \times (x^2 - 1) = x^2 - 1$$ Subtract remainder: $$ (x^2 - 1) - (x^2 - 1) = 0$$ 5. **Write the division result:** $$\frac{x^4 - 1}{x^2 - 1} = x^2 + 1$$ 6. **Factor and simplify:** Note that $$x^4 - 1$$ factors as a difference of squares: $$x^4 - 1 = (x^2 - 1)(x^2 + 1)$$ So the original expression simplifies directly to: $$\frac{(x^2 - 1)(x^2 + 1)}{x^2 - 1} = x^2 + 1$$ 7. **Conclusion:** The expression simplifies to $$x^2 + 1$$, which is a polynomial and does not require partial fraction decomposition. **Final answer:** $$x^2 + 1$$