To find which of the given options is a factor of the polynomial \( x^3 - 5x^2 - 2x + 24 \) while knowing that \( x - 3 \) is one of its linear factors, we can use polynomial long division or synthetic division.
First, we'll divide the polynomial \( x^3 - 5x^2 - 2x + 24 \) by \( x - 3 \).
Using Synthetic Division: We set up synthetic division with \( 3 \) (the root of \( x - 3 \)):
3 | 1 -5 -2 24
| 3 -6 -24
------------------
1 -2 -8 0
The result is:
- Quotient: \( 1x^2 - 2x - 8 \)
- Remainder: \( 0 \)
Thus, we have: \[ x^3 - 5x^2 - 2x + 24 = (x - 3)(x^2 - 2x - 8) \]
Now, we need to factor \( x^2 - 2x - 8 \).
To factor \( x^2 - 2x - 8 \):
- Find two numbers that multiply to \( -8 \) and add to \( -2 \). These numbers are \( -4 \) and \( 2 \).
So, we can factor it as: \[ x^2 - 2x - 8 = (x - 4)(x + 2) \]
Thus, the complete factorization of the polynomial is: \[ x^3 - 5x^2 - 2x + 24 = (x - 3)(x - 4)(x + 2) \]
Now, we can see that \( x + 2 \) and \( x - 4 \) are factors of the polynomial.
Therefore, the correct answer among your options is: \( x + 2 \).