To determine the greatest number of zeros that the polynomial \( f(x) = 7x^6 - 5x^5 + x \) can have, we can start by factoring the polynomial.
First, we can factor out the common term \( x \):
\[ f(x) = x(7x^5 - 5x^4 + 1) \]
This shows that \( x = 0 \) is one root, contributing one zero. Now we need to analyze the polynomial \( 7x^5 - 5x^4 + 1 \) to find the number of additional zeros it may have.
A polynomial of degree \( n \) can have at most \( n \) zeros. The polynomial \( 7x^5 - 5x^4 + 1 \) is a 5th degree polynomial, which means it can have up to 5 real roots (including complex roots).
Thus, taking into account the root \( x = 0 \) from the factored form, the total greatest number of zeros of \( f(x) \) is:
\[ 1 \text{ (for } x = 0\text{)} + 5 \text{ (from } 7x^5 - 5x^4 + 1\text{)} = 6 \]
Therefore, the greatest number of zeros that the polynomial \( f(x) = 7x^6 - 5x^5 + x \) can have is \( \boxed{6} \).