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The given function is:

\[
U(x) = 12x^5 + 6x^3 - 7x - 8
\]

The Rational Zeros Theorem provides a method to find all possible rational zeros (also known as roots) of a polynomial function by considering the factors of the constant term and the leading coefficient.

### Steps:
1. **Constant term**: The constant term in this polynomial is \( -8 \).
2. **Leading coefficient**: The leading coefficient (coefficient of \( x^5 \)) is \( 12 \).

The possible rational zeros are given by \( \frac{p}{q} \), where:
- \( p \) are the factors of the constant term \( -8 \).
- \( q \) are the factors of the leading coefficient \( 12 \).

### Factors of \( -8 \) (constant term):
\[
\pm 1, \pm 2, \pm 4, \pm 8
\]

### Factors of \( 12 \) (leading coefficient):
\[
\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12
\]

Now, using \( \frac{p}{q} \), we list all possible rational zeros by taking every combination of \( p \) divided by \( q \).

### Possible rational zeros:
\[
\pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{4}, \pm \frac{1}{6}, \pm \frac{1}{12}, \pm \frac{2}{3}, \pm \frac{2}{6}, \pm \frac{2}{12}, \pm \frac{4}{3}, \pm \frac{4}{6}, \pm \frac{4}{12}, \pm \frac{8}{3}, \pm \frac{8}{6}, \pm \frac{8}{12}
\]

Simplifying any fractions:

\[
\pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{4}, \pm \frac{1}{6}, \pm \frac{1}{12}, \pm \frac{2}{3}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm \frac{4}{3}, \pm \frac{2}{3}, \pm \frac{1}{3}, \pm \frac{8}{3}, \pm \frac{4}{3}, \pm \frac{2}{3}
\]

Thus, all possible rational zeros are:

\[
\pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{4}, \pm \frac{1}{6}, \pm \frac{1}{12}, \pm \frac{2}{3}, \pm \frac{4}{3}, \pm \frac{8}{3}
\]

Would you like further clarification or detailed verification of any of the roots?

Here are five related questions:
1. How can we test which of these possible zeros are actual zeros?
2. What is the significance of irrational or complex zeros in such polynomials?
3. Can synthetic division help in narrowing down the list of rational zeros?
4. What happens if none of the rational zeros work?
5. How does the degree of a polynomial affect the number of possible real zeros?

**Tip**: The Rational Zeros Theorem provides candidates for rational zeros, but you will still need to use substitution or other methods to check which values are actual roots.

List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see which actually are zeros). U(x) = 12x^5 + 6x^3 - 7x - 8

Discover all possible rational zeros for the polynomial U(x) = 12x^5 + 6x^3 - 7x - 8 using the Rational Zeros Theorem. This educational guide includes a comprehensive list and explanations suitable for high school students in Grades 10 to 12.

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