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Find the root(s) of [tex]f(x)=(x+5)^3(x-9)^2(x+1)[/tex]
A. -5 with multiplicity 3 B. 5 with multiplicity 3 C. -9 with multiplicity 2 D. 9 with multiplicity 2 E. -1 with multiplicity 0 F. -1 with multiplicity 1 G. 1 with multiplicity 0 H. 1 with multiplicity 1
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
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