F(x) = 4x^3+...+ 9
list all possible rational zeroes of the function

The Rational Root Theorem says any rational zeros of f must be a fraction with a factor of the constant (9 here) in the numerator and a factor of the leading coefficient (4 here) in the denominator.   We have to include negative factors.

Factors of 9:  -9, -3, -1, 1, 3, 9

Factors of 4: 1, 2, 4

We only have to consider negative numbers for the numerator to get all the possible values.

We just make the list, going through all the numerators one denominator at a time.

Possible rational zeros, i.e. possible rational values for x such that f(x)=0 are

-9, -3, -1, 1, 3, 9,   -9/2, -3/2, -1/2, 1/2, 3/2, 9/2,   -9/4, -3/4, -1/4, 1/4, 3/4, 9/4