The polynomial function given is $f(x) = x^3 + 3x^2 - 13x - 15$. The task is to find its linear factors.

Steps to solve:

1. Use Rational Root Theorem to find possible rational roots. This theorem suggests trying factors of the constant term (-15) divided by factors of the leading coefficient (1).

   Possible roots: $\pm 1, \pm 3, \pm 5, \pm 15$.

2. Check for roots by substituting these values into the polynomial. We need to find one root, and then factorize the polynomial further.

   • Trying $x = -1$: $f(-1) = (-1)^3 + 3(-1)^2 - 13(-1) - 15 = -1 + 3 + 13 - 15 = 0$ So, $x = -1$ is a root.

3. Divide the polynomial by $(x + 1)$ using synthetic or long division to get the quadratic factor.

   Performing synthetic division of $x^3 + 3x^2 - 13x - 15$ by $(x + 1)$ gives: $x^3 + 3x^2 - 13x - 15 = (x + 1)(x^2 + 2x - 15)$

4. Factor the quadratic $x^2 + 2x - 15$ further: $x^2 + 2x - 15 = (x + 5)(x - 3)$

Thus, the factorization of $f(x) = x^3 + 3x^2 - 13x - 15$ is: $f(x) = (x + 1)(x + 5)(x - 3)$

Correct Answer:

$f(x) = (x + 1)(x + 5)(x - 3)$

This corresponds to the fourth option in the image.

Would you like further details on any part of the process?

Here are some related questions:

1. How do we perform synthetic division?
2. What is the Rational Root Theorem?
3. How do we factor quadratic expressions?
4. Why do we need to check all possible roots?
5. What if none of the simple rational roots work?

Tip: When factoring polynomials, always start by checking simple integer roots using the Rational Root Theorem—it simplifies the problem significantly.