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Answer :
To find the zeros of the factored polynomial [tex]\( f(x) = (x-5)^5(x+1)^7 \)[/tex] and their multiplicities, we need to focus on the given factors of the polynomial.
1. Factor [tex]\((x - 5)^5\)[/tex]:
- This part of the polynomial tells us that one of the zeros is [tex]\( x = 5 \)[/tex].
- The exponent [tex]\( 5 \)[/tex] indicates the multiplicity of this zero. Therefore, [tex]\( x = 5 \)[/tex] has a multiplicity of 5.
2. Factor [tex]\((x + 1)^7\)[/tex]:
- This part shows that another zero is [tex]\( x = -1 \)[/tex].
- The exponent [tex]\( 7 \)[/tex] indicates the multiplicity of this zero. So, [tex]\( x = -1 \)[/tex] has a multiplicity of 7.
Based on this analysis, the zeros and their multiplicities for the polynomial [tex]\( f(x) = (x-5)^5(x+1)^7 \)[/tex] are:
- [tex]\( x = 5 \)[/tex] with multiplicity 5
- [tex]\( x = -1 \)[/tex] with multiplicity 7
The correct answer is:
[tex]\( x = -1 \)[/tex] with multiplicity 7, and [tex]\( x = 5 \)[/tex] with multiplicity 5.
1. Factor [tex]\((x - 5)^5\)[/tex]:
- This part of the polynomial tells us that one of the zeros is [tex]\( x = 5 \)[/tex].
- The exponent [tex]\( 5 \)[/tex] indicates the multiplicity of this zero. Therefore, [tex]\( x = 5 \)[/tex] has a multiplicity of 5.
2. Factor [tex]\((x + 1)^7\)[/tex]:
- This part shows that another zero is [tex]\( x = -1 \)[/tex].
- The exponent [tex]\( 7 \)[/tex] indicates the multiplicity of this zero. So, [tex]\( x = -1 \)[/tex] has a multiplicity of 7.
Based on this analysis, the zeros and their multiplicities for the polynomial [tex]\( f(x) = (x-5)^5(x+1)^7 \)[/tex] are:
- [tex]\( x = 5 \)[/tex] with multiplicity 5
- [tex]\( x = -1 \)[/tex] with multiplicity 7
The correct answer is:
[tex]\( x = -1 \)[/tex] with multiplicity 7, and [tex]\( x = 5 \)[/tex] with multiplicity 5.
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