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The question asks which of the given options is a factor of the polynomial \( x^3 - 5x^2 - 2x + 24 \), knowing that \( x - 3 \) is one of its linear factors.

### Steps to solve:

1. **Given:** One of the factors is \( x - 3 \), meaning that \( x = 3 \) is a root of the polynomial.
2. **Polynomial Division:** We can divide \( x^3 - 5x^2 - 2x + 24 \) by \( x - 3 \) using either synthetic or long division to find the quotient, which will help us identify the remaining factors.
   
Let's start by dividing \( x^3 - 5x^2 - 2x + 24 \) by \( x - 3 \).

### Polynomial division of \( x^3 - 5x^2 - 2x + 24 \) by \( x - 3 \):
 
- Divide the leading term \( x^3 \) by \( x \), which gives \( x^2 \).
- Multiply \( x^2 \) by \( x - 3 \), giving \( x^3 - 3x^2 \).
- Subtract \( (x^3 - 3x^2) \) from \( x^3 - 5x^2 \), which results in \( -2x^2 \).
- Divide \( -2x^2 \) by \( x \), which gives \( -2x \).
- Multiply \( -2x \) by \( x - 3 \), giving \( -2x^2 + 6x \).
- Subtract \( (-2x^2 + 6x) \) from \( -2x^2 - 2x \), which results in \( -8x \).
- Divide \( -8x \) by \( x \), which gives \( -8 \).
- Multiply \( -8 \) by \( x - 3 \), giving \( -8x + 24 \).
- Subtract \( (-8x + 24) \) from \( -8x + 24 \), resulting in 0.

So, the quotient is \( x^2 - 2x - 8 \).

### Factorization of \( x^2 - 2x - 8 \):
Now, we factor \( x^2 - 2x - 8 \). We need two numbers that multiply to give -8 and add to give -2. The numbers are 2 and -4.

Thus, \( x^2 - 2x - 8 \) factors as:
\[
(x - 4)(x + 2)
\]

### Conclusion:
The complete factorization of the polynomial \( x^3 - 5x^2 - 2x + 24 \) is:
\[
(x - 3)(x - 4)(x + 2)
\]

Hence, one of the factors is \( x + 2 \), which corresponds to the second option.

Let me know if you need more details, and here are some related questions to expand on this:

1. How do you perform synthetic division instead of long division?
2. Can all cubic polynomials be factored using this method?
3. How can you verify that \( x = 3 \) is a root of the polynomial?
4. What happens if a polynomial has no real roots?
5. How can you use the Rational Root Theorem to find possible roots?

**Tip:** When factoring cubic polynomials, checking for simple roots like integers using synthetic division or the Rational Root Theorem can speed up the process.

Which of the following is a factor of x^3 - 5x^2 - 2x + 24, given that x - 3 is one of the linear factors?

Discover how to factor the cubic polynomial x^3 - 5x^2 - 2x + 24 using polynomial division and the factor theorem. Learn step-by-step how to identify one factor and break down the remaining quadratic equation. Perfect for students in Grades 9-12 studying advanced algebra.

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