To solve this polynomial equation step-by-step and identify the correct factors from the given list, let's analyze each part of the equation and its factors.
Given:
[tex]\[ x^3 - 3x^2 - 10x + 24 = 0 \][/tex]
### Step-by-Step Solution:
1. Identify the polynomial:
The polynomial we are working with is [tex]\( x^3 - 3x^2 - 10x + 24 \)[/tex].
2. Factor the polynomial:
To factor the polynomial, we look for potential roots by trying various values for [tex]\( x \)[/tex] that might satisfy the equation [tex]\( x^3 - 3x^2 - 10x + 24 = 0 \)[/tex].
- Potential rational roots can be found using the Rational Root Theorem:
The theorem suggests potential roots as divisors of the constant term (24) over the leading coefficient (1). Therefore, test [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24 \)[/tex].
3. Testing potential roots:
- Testing [tex]\( x = 2 \)[/tex]:
[tex]\[
2^3 - 3(2)^2 - 10(2) + 24 = 8 - 12 - 20 + 24 = 0
\][/tex]
Since this evaluates to 0, [tex]\( x = 2 \)[/tex] is a root. Thus, [tex]\( (x - 2) \)[/tex] is a factor.
- Testing [tex]\( x = 4 \)[/tex]:
[tex]\[
4^3 - 3(4)^2 - 10(4) + 24 = 64 - 48 - 40 + 24 = 0
\][/tex]
Since this also evaluates to 0, [tex]\( x = 4 \)[/tex] is a root. Thus, [tex]\( (x - 4) \)[/tex] is a factor.
- Testing [tex]\( x = -3 \)[/tex]:
[tex]\[
(-3)^3 - 3(-3)^2 - 10(-3) + 24 = -27 - 27 + 30 + 24 = 0
\][/tex]
Since this also evaluates to 0, [tex]\( x = -3 \)[/tex] is a root. Thus, [tex]\( (x + 3) \)[/tex] is a factor.
4. Combine the factors:
Since [tex]\( x = 2, x = 4, \)[/tex] and [tex]\( x = -3 \)[/tex] are roots, the polynomial can be factored as:
[tex]\[
(x - 4)(x - 2)(x + 3)
\][/tex]
Therefore, the possible factors from the list are:
- [tex]\( (x - 4) \)[/tex] corresponds to option (d)
- [tex]\( (x - 2) \)[/tex] corresponds to option (d)
- [tex]\( (x + 3) \)[/tex] corresponds to option (f)
### Conclusion:
Based on the factorization of the given polynomial, the correct factors from the provided list are:
- (d) [tex]\( (x - 2) \)[/tex]
- (f) [tex]\( (x + 3) \)[/tex]
Thus, the selected factors are:
- (d) [tex]\( (x - 2) \)[/tex]
- (f) [tex]\( (x + 3) \)[/tex]
