To find the possible rational zeros of the cubic polynomial $x^3 + 8x^2 - 4x + 8$, we can apply the Rational Root Theorem. This theorem states that any possible rational root of a polynomial with integer coefficients will be of the form:

$\frac{p}{q}$ where:

• $p$ is a factor of the constant term (in this case, $8$),
• $q$ is a factor of the leading coefficient (in this case, $1$, since the leading term is $x^3$).

Step 1: Identify factors of the constant term (8):

The factors of 8 are: $\pm 1, \pm 2, \pm 4, \pm 8$

Step 2: Identify factors of the leading coefficient (1):

The only factors of 1 are: $\pm 1$

Step 3: List all possible rational zeros:

Since $q$ can only be $\pm 1$, the possible rational roots are just the factors of 8, which are: $\pm 1, \pm 2, \pm 4, \pm 8$

Thus, the possible rational zeros of the polynomial $x^3 + 8x^2 - 4x + 8$ are: $\boxed{\pm 1, \pm 2, \pm 4, \pm 8}$

Would you like further details on how to verify if any of these roots are actual solutions? Or any questions about this process?

Related Questions:

1. How can we test which of the possible rational roots are actual zeros?
2. What is the Rational Root Theorem and how is it used?
3. Can we apply synthetic division to verify the roots?
4. What would happen if the leading coefficient were not 1?
5. How do irrational or complex roots come into play in higher degree polynomials?

Tip: Always double-check potential rational roots by substituting them into the polynomial!