To list all possible rational zeros, we can use the Rational Root Theorem. This theorem states that any potential rational zero of a polynomial $P(x)$ is of the form:

$\frac{p}{q}$

where:

• $p$ is a factor of the constant term (the term with no $x$).
• $q$ is a factor of the leading coefficient (the coefficient of the highest power of $x$).

1) $P(x) = x^3 + 2x^2 - x - 2$

• Constant term (p): -2 → Factors are $\pm 1, \pm 2$
• Leading coefficient (q): 1 → Factors are $\pm 1$

The possible rational zeros are:

$\frac{p}{q} = \pm 1, \pm 2$

5) $P(x) = 4x^4 - 8x^3 - x^2 + 8x - 8$

• Constant term (p): -8 → Factors are $\pm 1, \pm 2, \pm 4, \pm 8$
• Leading coefficient (q): 4 → Factors are $\pm 1, \pm 2, \pm 4$

The possible rational zeros are:

$\frac{p}{q} = \pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}, \pm \frac{1}{4}$

These lists represent the possible rational solutions. You would need to test each value to determine which are actual roots by substituting them into the polynomial or using synthetic division.

Would you like me to check any of these potential roots step-by-step?

Here are 5 related questions to expand on this:

1. How can we use synthetic division to confirm if a rational root works?
2. What happens if no rational zeros are found after testing all possibilities?
3. How do we factor polynomials if we know one root?
4. What are the differences between real and complex roots?
5. How can graphing a polynomial help us identify approximate roots?

Tip: If testing roots manually, try starting with the smallest integers to save time.