To find the polynomial function \( f(x) \) with the specified roots and multiplicities, we must consider each root and their effects on the polynomial.
1. The roots given are \( -4 \), \( 2 \), and \( 9 \) each with multiplicity 1.
2. There is also a root at \( -5 \) with multiplicity 3.
Let's build the polynomial step by step:
### Step-by-step process:
1. For the root \( -4 \) with multiplicity 1, the factor corresponding to this root is \( (x + 4) \).
2. For the root \( 2 \) with multiplicity 1, the factor corresponding to this root is \( (x - 2) \).
3. For the root \( 9 \) with multiplicity 1, the factor corresponding to this root is \( (x - 9) \).
4. For the root \( -5 \) with multiplicity 3, the factor corresponding to this root is \( (x + 5)^3 \).
When combining these factors, we get the polynomial with the correct roots and multiplicities. It will look like:
[tex]\[ f(x) = (x + 5)^3 (x + 4) (x - 2) (x - 9) \][/tex]
### Leading coefficient:
- The leading coefficient is \( 1 \), as specified in the problem. This means that we do not need to multiply the entire polynomial by any constant other than \( 1 \).
### Final Polynomial:
Combining all these factors, the polynomial that satisfies all the conditions is:
[tex]\[ f(x) = (x + 5)(x + 5)(x + 5)(x + 4)(x - 2)(x - 9) \][/tex]
So, the correct polynomial function is:
[tex]\[ f(x) = (x + 5)(x + 5)(x + 5)(x + 4)(x - 2)(x - 9) \][/tex]
Which matches the option:
[tex]\[ \boxed{f(x) = (x+5)(x+5)(x+5)(x+4)(x-2)(x-9)} \][/tex]
