To write the partial fraction decomposition for the given function:

$f(x) = \frac{x^2 - 3}{(x - 5)(x + 5)^3},$

we follow these steps:

Step 1: Identify the Denominator Structure

The denominator has the following factors:

1. A linear term $(x - 5)$,
2. A cubic term $(x + 5)^3$.

Step 2: Set Up the Partial Fractions

For the term $(x - 5)$, we assign a constant $A$.
For the term $(x + 5)^3$, we decompose it into three terms: $\frac{B}{x + 5}, \quad \frac{C}{(x + 5)^2}, \quad \frac{D}{(x + 5)^3}.$

Thus, the general decomposition is: $\frac{x^2 - 3}{(x - 5)(x + 5)^3} = \frac{A}{x - 5} + \frac{B}{x + 5} + \frac{C}{(x + 5)^2} + \frac{D}{(x + 5)^3}.$

Step 3: Write the Complete Form

The full partial fraction decomposition is: $f(x) = \frac{A}{x - 5} + \frac{B}{x + 5} + \frac{C}{(x + 5)^2} + \frac{D}{(x + 5)^3}.$

Since the problem specifies "without evaluating the constants," we stop here.

Would you like details on how to solve for the constants $A, B, C, D$, or do you have additional questions?

Related Questions:

1. How do we find the constants $A, B, C, D$ using algebraic techniques?
2. What happens if the numerator is of higher degree than the denominator?
3. How can this method extend to cases with repeated linear and quadratic factors?
4. Can partial fraction decomposition be applied to improper rational functions?
5. How can partial fractions be used to integrate rational functions?

Tip:

When setting up partial fractions, always check the degrees of the numerator and denominator first. If the numerator's degree is greater than or equal to the denominator's, perform polynomial long division first!