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?(1 point)
zeros at most

Answer 1

The degree of a polynomial corresponds to the highest power of the variable in the polynomial expression. In the given polynomial \( f(x) = (x-1)^3 (x+7) \), we can find the degree by calculating the contributions from each factor:

1. The factor \((x-1)^3\) has a degree of 3.
2. The factor \((x+7)\) has a degree of 1.

To find the total degree of the polynomial, we sum the degrees of the individual factors:

\[
\text{Degree of } f(x) = 3 + 1 = 4
\]

According to the Fundamental Theorem of Algebra, a polynomial of degree \( n \) can have at most \( n \) complex zeros (including real and repeated zeros). Since we established that the degree of \( f(x) \) is 4, the greatest number of zeros it could have is:

\[
\text{Greatest number of zeros} = 4
\]

So, the polynomial \( f(x) = (x-1)^3(x+7) \) could have at most **4 zeros**.