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Based on the degree of the polynomial f(x)=(x−1)^3(x+7), what is the greatest number of zeros it could have?Question
Based on the degree of the polynomial f(x)=(x-1)^3(x+7) what is the greatest number of zeros it could have?
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GPT-4o mini
The degree of the polynomial \( f(x) = (x - 1)^3 (x + 7) \) can be determined by looking at the factors.
1. The factor \( (x - 1)^3 \) has a degree of 3.
2. The factor \( (x + 7) \) has a degree of 1.
To find the degree of the polynomial \( f(x) \), we add the degrees of the individual factors:
\[
\text{Degree of } f(x) = 3 + 1 = 4.
\]
The greatest number of zeros a polynomial can have is equal to its degree. Therefore, since the degree of \( f(x) \) is 4, the greatest number of zeros it could have is:
\[
\boxed{4}.
\]
It's worth noting that the zeros of the polynomial are not all distinct; \( x = 1 \) is a zero with a multiplicity of 3, and \( x = -7 \) is a zero with a multiplicity of 1. However, the total count of zeros, considering their multiplicities, still adds up to 4.
1. The factor \( (x - 1)^3 \) has a degree of 3.
2. The factor \( (x + 7) \) has a degree of 1.
To find the degree of the polynomial \( f(x) \), we add the degrees of the individual factors:
\[
\text{Degree of } f(x) = 3 + 1 = 4.
\]
The greatest number of zeros a polynomial can have is equal to its degree. Therefore, since the degree of \( f(x) \) is 4, the greatest number of zeros it could have is:
\[
\boxed{4}.
\]
It's worth noting that the zeros of the polynomial are not all distinct; \( x = 1 \) is a zero with a multiplicity of 3, and \( x = -7 \) is a zero with a multiplicity of 1. However, the total count of zeros, considering their multiplicities, still adds up to 4.
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