The problem is asking how many second-type partial fractions will be present in the formal decomposition of the expression:

$\frac{x^2 - 5}{x^3 + x^2 - x - 1}$

Step 1: Factor the Denominator

To perform partial fraction decomposition, we first factor the denominator $x^3 + x^2 - x - 1$.

1. Factor by grouping: $x^3 + x^2 - x - 1 = x^2(x + 1) - 1(x + 1) = (x^2 - 1)(x + 1)$
2. Factor $x^2 - 1$ as a difference of squares: $x^2 - 1 = (x - 1)(x + 1)$
3. Thus, the denominator factors as: $x^3 + x^2 - x - 1 = (x - 1)(x + 1)^2$

Step 2: Set Up Partial Fraction Decomposition

The partial fraction decomposition will involve terms based on the factors of the denominator. For each linear factor $(x - 1)$ and $(x + 1)^2$, we set up terms as follows:

1. For the factor $(x - 1)$: a single fraction term $\frac{A}{x - 1}$.
2. For the repeated factor $(x + 1)^2$: two terms are required, $\frac{B}{x + 1}$ and $\frac{C}{(x + 1)^2}$.

Thus, the decomposition is:

$\frac{x^2 - 5}{(x - 1)(x + 1)^2} = \frac{A}{x - 1} + \frac{B}{x + 1} + \frac{C}{(x + 1)^2}$

Step 3: Identify Second-Type Partial Fractions

In this context, "second-type partial fractions" refers to fractions with terms of higher powers in the denominator. Here, $\frac{C}{(x + 1)^2}$ is the only term with a higher power (specifically, $(x + 1)^2$).

Conclusion

There is 1 second-type partial fraction in the decomposition.

Would you like a step-by-step breakdown of how to solve for each coefficient (A, B, and C)?

Here are five related questions:

1. How do we solve for the coefficients $A$, $B$, and $C$ in partial fractions?
2. What are the types of partial fraction decompositions for different polynomial degrees in the denominator?
3. How would the decomposition change if the denominator included a quadratic irreducible factor?
4. What are the general rules for setting up partial fractions with repeated factors in the denominator?
5. Can partial fractions be used in integration, and how does the decomposition simplify the process?

Tip: When factoring the denominator, always look for common patterns such as grouping or difference of squares to simplify the process.