To determine the greatest number of zeros the polynomial \( f(x) = 7x^6 - 5x^5 + x \) could have, we can analyze its degree and the possible number of roots.
- The degree of the polynomial is determined by the highest power of \( x \), which is \( 6 \) in \( 7x^6 \).
- A polynomial of degree \( n \) can have at most \( n \) real roots.
Since the degree of this polynomial is \( 6 \), it can have up to \( 6 \) real zeros.
To confirm whether \( f(x) \) can achieve the maximum of \( 6 \) real zeros, we can check if it is possible for the polynomial to have \( 6 \) distinct real zeros by establishing changes in sign, or by finding actual roots.
However, regarding the question, the answer is:
\[ \text{The greatest number of zeros the polynomial } f(x) \text{ could have is } 6. \]
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