To find the roots of the polynomial function [tex]\(f(x) = (x + 5)^3 (x - 9)^2 (x + 1)\)[/tex], we must identify the values of [tex]\(x\)[/tex] that make [tex]\(f(x) = 0\)[/tex]. The roots of the function can be found by setting each factor of the polynomial to zero and solving for [tex]\(x\)[/tex].
Let's break down the polynomial into its factors and find the roots and their corresponding multiplicities:
1. Factor [tex]\( (x + 5) \)[/tex]
- To find the root from this factor, set [tex]\( x + 5 = 0 \)[/tex].
- Solving for [tex]\( x \)[/tex], we get [tex]\( x = -5 \)[/tex].
- The factor [tex]\( (x + 5) \)[/tex] is raised to the power of 3.
- Therefore, [tex]\( x = -5 \)[/tex] is a root with multiplicity 3.
2. Factor [tex]\( (x - 9) \)[/tex]
- To find the root from this factor, set [tex]\( x - 9 = 0 \)[/tex].
- Solving for [tex]\( x \)[/tex], we get [tex]\( x = 9 \)[/tex].
- The factor [tex]\( (x - 9) \)[/tex] is raised to the power of 2.
- Therefore, [tex]\( x = 9 \)[/tex] is a root with multiplicity 2.
3. Factor [tex]\( (x + 1) \)[/tex]
- To find the root from this factor, set [tex]\( x + 1 = 0 \)[/tex].
- Solving for [tex]\( x \)[/tex], we get [tex]\( x = -1 \)[/tex].
- The factor [tex]\( (x + 1) \)[/tex] is raised to the power of 1.
- Therefore, [tex]\( x = -1 \)[/tex] is a root with multiplicity 1.
Now, let's summarize the roots and their multiplicities:
- [tex]\( x = -5 \)[/tex] with multiplicity 3.
- [tex]\( x = 9 \)[/tex] with multiplicity 2.
- [tex]\( x = -1 \)[/tex] with multiplicity 1.
Given the choices, the correct responses are:
- [tex]\( -5 \)[/tex] with multiplicity 3
- [tex]\( 9 \)[/tex] with multiplicity 2
- [tex]\( -1 \)[/tex] with multiplicity 1
