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

Answer: - 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