Based on the degree of the polynomial f(x)=(x-1)^3(x+7) what is the greatest number of zeros it could have?

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.