To determine which polynomial function [tex]\( f(x) \)[/tex] has the specified roots and multiplicities, we need to ensure the polynomial meets the given criteria:
- Leading coefficient of 1
- Roots [tex]\(-4, 2\)[/tex], and [tex]\(9\)[/tex] with multiplicity 1
- Root [tex]\(-5\)[/tex] with multiplicity 3
We'll construct the polynomial step-by-step:
1. Identify the factors for each root:
- A root [tex]\( r \)[/tex] of a polynomial translates to a factor of [tex]\( (x - r) \)[/tex].
- 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].
2. Combine all factors to form the polynomial:
[tex]\[
f(x) = (x + 5)^3 (x + 4) (x - 2) (x - 9)
\][/tex]
3. Check the leading coefficient:
- The polynomial [tex]\( f(x) = (x + 5)^3 (x + 4) (x - 2) (x - 9) \)[/tex] has a leading coefficient of 1, since all coefficients involve a basic unscaled binomial [tex]\( (x - r) \)[/tex].
4. Match given options with our polynomial:
- Option 1: [tex]\(3(x+5)(x+4)(x-2)(x-9)\)[/tex]
- This polynomial does not match as it scales the factors by 3 and does not have [tex]\( (x+5)^3 \)[/tex].
- Option 2: [tex]\(3(x-5)(x-4)(x+2)(x+9)\)[/tex]
- This polynomial has entirely different roots and does not match.
- Option 3: [tex]\((x+5)(x+5)(x+5)(x+4)(x-2)(x-9)\)[/tex]
- This polynomial matches perfectly with [tex]\((x+5)^3(x+4)(x-2)(x-9)\)[/tex].
- Option 4: [tex]\((x-5)(x-5)(x-5)(x-4)(x+2)(x+9)\)[/tex]
- This polynomial has entirely different roots and does not match.
Since Option 3 is the one that matches our constructed polynomial, the answer is:
[tex]\[ \boxed{(x+5)(x+5)(x+5)(x+4)(x-2)(x-9)} \][/tex]
