To find the greatest number of zeros a polynomial can have, we can use the fact that the degree of the polynomial gives us the maximum number of distinct real roots.
The polynomial in question is
\[ f(x) = 7x^6 - 5x^5 + x. \]
The degree of this polynomial is 6, which means that it can have at most 6 roots (counting multiplicities).
To confirm, we can also note that:
- A polynomial of degree \( n \) can have at most \( n \) distinct roots.
- Roots can be real or complex, but the maximum number of roots (including complex) is still equal to the degree of the polynomial.
Since \( f(x) \) has a degree of 6, the greatest number of zeros (real or complex) that the polynomial can have is:
\[ \text{Greatest number of zeros} = 6. \]
Thus, the answer is that the polynomial \( f(x) \) can have a maximum of 6 zeros.