To determine which polynomial function [tex]\( f(x) \)[/tex] matches the given conditions, we need to consider the roots and their multiplicities, as well as the leading coefficient of 1.
### Given Information:
- Roots: [tex]\(-4, 2, 9\)[/tex] each with multiplicity 1
- Root: [tex]\(-5\)[/tex] with multiplicity 3
- Leading coefficient is 1
### Step-by-Step Solution:
1. Identify the Factor Structure:
- Roots with multiplicity define the linear factors of the polynomial.
- For a root [tex]\( r \)[/tex] with multiplicity [tex]\( m \)[/tex], the factor [tex]\((x - r)^m\)[/tex] is included.
2. Construct the Polynomial:
- For the root [tex]\(-4\)[/tex] with multiplicity 1: the factor is [tex]\((x + 4)\)[/tex]
- For the root [tex]\(2\)[/tex] with multiplicity 1: the factor is [tex]\((x - 2)\)[/tex]
- For the root [tex]\(9\)[/tex] with multiplicity 1: the factor is [tex]\((x - 9)\)[/tex]
- For the root [tex]\(-5\)[/tex] with multiplicity 3: the factor is [tex]\((x + 5)^3\)[/tex]
3. Form the Polynomial Expression:
- Combine these factors to form the polynomial, taking care to include the multiplicity correctly.
- Therefore, the polynomial [tex]\( f(x) \)[/tex] can be written as:
[tex]\[
f(x) = (x + 5)^3 (x + 4) (x - 2) (x - 9)
\][/tex]
4. Verify the Leading Coefficient:
- Since we are told the leading coefficient should be 1, we must ensure no extra coefficients are multiplied that will alter this. Multiplying the individual linear factors, we verify that the polynomial formed, indeed, maintains the leading coefficient of 1.
### Conclusion:
Given the structure and requirements, the polynomial function that satisfies all these conditions is:
[tex]\[
f(x) = (x+5)(x+5)(x+5)(x+4)(x-2)(x-9)
\][/tex]
Thereby, the correct option among the ones provided is:
[tex]\[
f(x) = (x+5)(x+5)(x+5)(x+4)(x-2)(x-9)
\][/tex]
This corresponds to the third listed option in the given choices:
[tex]\[
\boxed{3}
\][/tex]
