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?
Answers
GPT-4o mini
To determine the greatest number of zeros a polynomial can have, we need to identify the degree of the polynomial.
Let's analyze the given polynomial \( f(x) = (x - 1)^3 (x + 7) \).
1. The term \( (x - 1)^3 \) has a degree of 3 because the exponent is 3.
2. The term \( (x + 7) \) has a degree of 1 because it is a linear factor.
To find the total degree of \( f(x) \), we add the degrees of each of the factors:
\[
\text{Degree of } f(x) = 3 + 1 = 4.
\]
A polynomial of degree \( n \) can have at most \( n \) distinct roots (or zeros). Thus, since our polynomial \( f(x) \) has a degree of 4, it can have at most 4 zeros.
Now, we need to consider multiplicities of the zeros. The zero from the factor \( (x - 1)^3 \) is \( x = 1 \) which has a multiplicity of 3, and the zero from the factor \( (x + 7) \) is \( x = -7 \) which has a multiplicity of 1.
Therefore, while we have multiplicities, they don’t change the count of distinct zeros. Hence, the number of distinct zeros is:
- \( x = 1 \) (multiplicity 3),
- \( x = -7 \) (multiplicity 1).
However, since we are asked about the greatest number of zeros, including multiplicities, the total count remains:
\[
\text{Total zeros (counting multiplicities)} = 3 + 1 = 4.
\]
Thus, the greatest number of zeros \( f(x) \) could have is
\[
\boxed{4}.
\]
Let's analyze the given polynomial \( f(x) = (x - 1)^3 (x + 7) \).
1. The term \( (x - 1)^3 \) has a degree of 3 because the exponent is 3.
2. The term \( (x + 7) \) has a degree of 1 because it is a linear factor.
To find the total degree of \( f(x) \), we add the degrees of each of the factors:
\[
\text{Degree of } f(x) = 3 + 1 = 4.
\]
A polynomial of degree \( n \) can have at most \( n \) distinct roots (or zeros). Thus, since our polynomial \( f(x) \) has a degree of 4, it can have at most 4 zeros.
Now, we need to consider multiplicities of the zeros. The zero from the factor \( (x - 1)^3 \) is \( x = 1 \) which has a multiplicity of 3, and the zero from the factor \( (x + 7) \) is \( x = -7 \) which has a multiplicity of 1.
Therefore, while we have multiplicities, they don’t change the count of distinct zeros. Hence, the number of distinct zeros is:
- \( x = 1 \) (multiplicity 3),
- \( x = -7 \) (multiplicity 1).
However, since we are asked about the greatest number of zeros, including multiplicities, the total count remains:
\[
\text{Total zeros (counting multiplicities)} = 3 + 1 = 4.
\]
Thus, the greatest number of zeros \( f(x) \) could have is
\[
\boxed{4}.
\]