To find the polynomial function [tex]\( f(x) \)[/tex] with the given properties, we need to construct it step-by-step based on the roots and their multiplicities.
1. Leading Coefficient: The polynomial should have a leading coefficient of 1. This means the highest-degree term when the polynomial is expanded will have a coefficient of 1.
2. Roots and Multiplicities:
- Root [tex]\( -4 \)[/tex] with multiplicity 1 means there is a factor [tex]\( (x + 4) \)[/tex].
- Root [tex]\( 2 \)[/tex] with multiplicity 1 means there is a factor [tex]\( (x - 2) \)[/tex].
- Root [tex]\( 9 \)[/tex] with multiplicity 1 means there is a factor [tex]\( (x - 9) \)[/tex].
- Root [tex]\( -5 \)[/tex] with multiplicity 3 means there is a factor [tex]\( (x + 5)^3 \)[/tex].
3. Constructing the Polynomial: Combine all these factors to form the polynomial:
[tex]\[
f(x) = (x + 5)^3 (x + 4) (x - 2) (x - 9)
\][/tex]
Now we will compare this polynomial to the given options:
Option 1: [tex]\( f(x) = 3(x + 5)(x + 4)(x - 2)(x - 9) \)[/tex]
- This doesn't match because it has a multiplicative factor of 3 and [tex]\( (x + 5) \)[/tex] is not cubed.
Option 2: [tex]\( f(x) = 3(x - 5)(x - 4)(x + 2)(x + 9) \)[/tex]
- This is incorrect as it has different roots and a leading coefficient of 3, not 1.
Option 3: [tex]\( f(x) = (x + 5)(x + 5)(x + 5)(x + 4)(x - 2)(x - 9) \)[/tex]
- This matches the required polynomial because the factor [tex]\( (x + 5) \)[/tex] is cubed, and it includes all the other correct factors correctly.
Option 4: [tex]\( f(x) = (x - 5)(x - 5)(x - 5)(x - 4)(x + 2)(x + 9) \)[/tex]
- This is incorrect because it has different roots: [tex]\( x = 5 \)[/tex], [tex]\( -4 \)[/tex], [tex]\( -2 \)[/tex], [tex]\( -9 \)[/tex].
Therefore, the correct polynomial based on the given properties is:
[tex]\[ f(x) = (x+5)(x+5)(x+5)(x+4)(x-2)(x-9) \][/tex]
Thus, the correct option is:
[tex]\[ \boxed{3} \][/tex]
