To find a polynomial function [tex]\( f(x) \)[/tex] with a leading coefficient of 1 that has roots at [tex]\( -4 \)[/tex], [tex]\( 2 \)[/tex], and [tex]\( 9 \)[/tex] each with multiplicity 1, and a root at [tex]\( -5 \)[/tex] with multiplicity 3, follow these steps:
### Step-by-Step Solution:
1. Identify the roots and multiplicities:
- Root [tex]\( -4 \)[/tex] with multiplicity 1
- Root [tex]\( 2 \)[/tex] with multiplicity 1
- Root [tex]\( 9 \)[/tex] with multiplicity 1
- Root [tex]\( -5 \)[/tex] with multiplicity 3
2. Construct factors for each root:
- The factor for root [tex]\( -4 \)[/tex] is [tex]\( (x + 4) \)[/tex].
- The factor for root [tex]\( 2 \)[/tex] is [tex]\( (x - 2) \)[/tex].
- The factor for root [tex]\( 9 \)[/tex] is [tex]\( (x - 9) \)[/tex].
- The factor for root [tex]\( -5 \)[/tex] with multiplicity 3 is [tex]\( (x + 5)^3 \)[/tex].
3. Form the polynomial by multiplying the factors:
[tex]\[
f(x) = (x + 5)^3 \cdot (x + 4) \cdot (x - 2) \cdot (x - 9)
\][/tex]
4. Check the given options:
- First option: [tex]\( f(x) = 3(x + 5)(x + 4)(x - 2)(x - 9) \)[/tex]
- Second option: [tex]\( f(x) = 3(x - 5)(x - 4)(x + 2)(x + 9) \)[/tex]
- Third option: [tex]\( f(x) = (x + 5)(x + 5)(x + 5)(x + 4)(x - 2)(x - 9) \)[/tex]
- Fourth option: [tex]\( f(x) = (x - 5)(x - 5)(x - 5)(x - 4)(x + 2)(x + 9) \)[/tex]
From the options given, the third option matches the derived polynomial function:
[tex]\[
f(x) = (x + 5)^3 \cdot (x + 4) \cdot (x - 2) \cdot (x - 9)
\][/tex]
Therefore, the correct polynomial function is given in the third choice:
[tex]\[
f(x) = (x + 5)(x + 5)(x + 5)(x + 4)(x - 2)(x - 9)
\][/tex]
The answer is 3.
