Final answer:
The possible rational zeros of the polynomial x^5 - 6 are ±1, ±2, ±3, and ±6. None of these possible zeros are actual zeros of the polynomial.
Step-by-step explanation:
The possible rational zeros of the polynomial x5 - 6 can be found using the Rational Root Theorem. According to this theorem, any rational zero of the polynomial must be of the form p/q, where p is a factor of the constant term (-6) and q is a factor of the leading coefficient (1).
In this case, the factors of -6 are -1, -2, -3, 1, 2, 3, 6. The factors of 1 are 1 and -1.
Therefore, the possible rational zeros of the polynomial are ±1, ±2, ±3, and ±6.
We can verify these possible zeros by substituting them into the polynomial. For example, if we substitute x = 1, we get 15 - 6 = 1 - 6 = -5, which is not equal to zero.
Similarly, if we substitute x = -1, we get (-1)5 - 6 = -1 - 6 = -7, which is also not equal to zero.
Therefore, none of the possible rational zeros are actual zeros of the polynomial.