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Answer :
To find the zeros of the polynomial [tex]\( f(x) = (x-5)^5(x+1)^7 \)[/tex] and their multiplicities, we need to look at each factor of the polynomial individually.
1. Identify the factors:
The polynomial is given as a product of two factors: [tex]\((x-5)^5\)[/tex] and [tex]\((x+1)^7\)[/tex].
2. Find the zeros from each factor:
- From the factor [tex]\((x-5)^5\)[/tex], we set [tex]\(x-5 = 0\)[/tex]. Solving this gives [tex]\(x = 5\)[/tex]. The exponent 5 indicates the multiplicity of this zero, which is 5.
- From the factor [tex]\((x+1)^7\)[/tex], we set [tex]\(x+1 = 0\)[/tex]. Solving this gives [tex]\(x = -1\)[/tex]. The exponent 7 indicates the multiplicity of this zero, which is 7.
3. Write the zeros with their multiplicities:
- Zero: [tex]\(x = 5\)[/tex]; Multiplicity: 5
- Zero: [tex]\(x = -1\)[/tex]; Multiplicity: 7
Based on this breakdown, the correct answer is:
- [tex]\(x = -1\)[/tex] with multiplicity 7
- [tex]\(x = 5\)[/tex] with multiplicity 5
This aligns with the selection: [tex]\(x=-1\)[/tex] with multiplicity 7, and [tex]\(x=5\)[/tex] with multiplicity 5.
1. Identify the factors:
The polynomial is given as a product of two factors: [tex]\((x-5)^5\)[/tex] and [tex]\((x+1)^7\)[/tex].
2. Find the zeros from each factor:
- From the factor [tex]\((x-5)^5\)[/tex], we set [tex]\(x-5 = 0\)[/tex]. Solving this gives [tex]\(x = 5\)[/tex]. The exponent 5 indicates the multiplicity of this zero, which is 5.
- From the factor [tex]\((x+1)^7\)[/tex], we set [tex]\(x+1 = 0\)[/tex]. Solving this gives [tex]\(x = -1\)[/tex]. The exponent 7 indicates the multiplicity of this zero, which is 7.
3. Write the zeros with their multiplicities:
- Zero: [tex]\(x = 5\)[/tex]; Multiplicity: 5
- Zero: [tex]\(x = -1\)[/tex]; Multiplicity: 7
Based on this breakdown, the correct answer is:
- [tex]\(x = -1\)[/tex] with multiplicity 7
- [tex]\(x = 5\)[/tex] with multiplicity 5
This aligns with the selection: [tex]\(x=-1\)[/tex] with multiplicity 7, and [tex]\(x=5\)[/tex] with multiplicity 5.
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