To find the roots of the polynomial [tex]\( f(x) = (x+5)^3(x-9)^2(x+1) \)[/tex] and their multiplicities, we proceed as follows:
### Step-by-Step Solution
1. Identify the factors of the polynomial: The polynomial [tex]\( f(x) = (x+5)^3(x-9)^2(x+1) \)[/tex] is already factored, so we can directly analyze the factors.
2. Determine the roots from the factors:
- The term [tex]\((x+5)^3\)[/tex] gives a root where [tex]\(x + 5 = 0\)[/tex]. Solving this, we find:
[tex]\[
x = -5
\][/tex]
- The term [tex]\((x-9)^2\)[/tex] gives a root where [tex]\(x - 9 = 0\)[/tex]. Solving this, we find:
[tex]\[
x = 9
\][/tex]
- The term [tex]\((x+1)\)[/tex] gives a root where [tex]\(x + 1 = 0\)[/tex]. Solving this, we find:
[tex]\[
x = -1
\][/tex]
3. Determine the multiplicities of the roots:
- The factor [tex]\((x+5)^3\)[/tex] indicates that the root [tex]\( x = -5 \)[/tex] has a multiplicity of 3 because it is raised to the power of 3.
- The factor [tex]\((x-9)^2\)[/tex] indicates that the root [tex]\( x = 9 \)[/tex] has a multiplicity of 2 because it is raised to the power of 2.
- The factor [tex]\((x+1)\)[/tex] indicates that the root [tex]\( x = -1 \)[/tex] has a multiplicity of 1 because it is raised to the power of 1.
### Conclusion
Based on the analysis of the polynomial, the roots and their respective multiplicities are:
- [tex]\( x = -5 \)[/tex] with multiplicity 3
- [tex]\( x = 9 \)[/tex] with multiplicity 2
- [tex]\( x = -1 \)[/tex] with multiplicity 1
Therefore, the correct answers are:
- [tex]\(-5\)[/tex] with multiplicity 3
- [tex]\(9\)[/tex] with multiplicity 2
- [tex]\(-1\)[/tex] with multiplicity 1
