To find the roots and their multiplicities for the function [tex]\( f(x) = (x+5)^3 (x-9)^2 (x+1) \)[/tex], let's analyze the expression step by step:
1. Understand the factors:
[tex]\[
f(x) = (x+5)^3 (x-9)^2 (x+1)
\][/tex]
This polynomial is already factored, which makes it easier to identify the roots and their multiplicities. Each factor of the form [tex]\((x-a)^n\)[/tex] tells us that [tex]\(x = a\)[/tex] is a root with multiplicity [tex]\(n\)[/tex].
2. Identify the roots and their multiplicities:
- The factor [tex]\((x+5)^3\)[/tex] indicates a root at [tex]\(x = -5\)[/tex] with multiplicity 3.
- The factor [tex]\((x-9)^2\)[/tex] indicates a root at [tex]\(x = 9\)[/tex] with multiplicity 2.
- The factor [tex]\((x+1)\)[/tex] indicates a root at [tex]\(x = -1\)[/tex] with multiplicity 1.
3. Summarize the results:
- Root: [tex]\( -5 \)[/tex], Multiplicity: 3
- Root: [tex]\( 9 \)[/tex], Multiplicity: 2
- Root: [tex]\( -1 \)[/tex], Multiplicity: 1
So, the roots and their multiplicities for the function [tex]\( f(x) = (x+5)^3 (x-9)^2 (x+1) \)[/tex] are:
- [tex]\(-5\)[/tex] with multiplicity 3
- [tex]\(9\)[/tex] with multiplicity 2
- [tex]\(-1\)[/tex] with multiplicity 1
To match these findings with the provided options:
- [tex]\(-5\)[/tex] with multiplicity 3 — Correct.
- [tex]\(5\)[/tex] with multiplicity 3 — Incorrect, as [tex]\( f(x)=0 \)[/tex] has no root at [tex]\(x=5\)[/tex].
- [tex]\(-9\)[/tex] with multiplicity 2 — Incorrect, the correct root is [tex]\(9\)[/tex] with multiplicity 2.
- [tex]\(9\)[/tex] with multiplicity 2 — Correct.
- [tex]\(-1\)[/tex] with multiplicity 0 — Incorrect, [tex]\(x = -1\)[/tex] is indeed a root with nonzero multiplicity.
- [tex]\(-1\)[/tex] with multiplicity 1 — Correct.
- [tex]\(1\)[/tex] with multiplicity 0 — Correct, as [tex]\( f(x)=0 \)[/tex] has no root at [tex]\(x=1\)[/tex].
- [tex]\(1\)[/tex] with multiplicity 1 — Incorrect, as [tex]\( f(x)=0 \)[/tex] has no root at [tex]\(x=1\)[/tex].
Hence the correct answers are:
[tex]\(\boxed{-5 \text{ with multiplicity 3}}\)[/tex]
[tex]\(\boxed{9 \text{ with multiplicity 2}}\)[/tex]
[tex]\(\boxed{-1 \text{ with multiplicity 1}}\)[/tex]
