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The equation provided is:

\[
x^3 - 2x^2 - 9x + 18 = 0
\]

### Part (a): List all rational roots according to the Rational Zero Theorem.

The **Rational Zero Theorem** states that any possible rational root of a polynomial is of the form:

\[
\frac{p}{q}
\]

where:
- \(p\) is a factor of the constant term (in this case, \(18\)),
- \(q\) is a factor of the leading coefficient (in this case, \(1\), the coefficient of \(x^3\)).

#### Factors of the constant term 18:
\[
p = \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18
\]

#### Factors of the leading coefficient 1:
\[
q = \pm 1
\]

Thus, the possible rational roots are given by all the combinations of \(\frac{p}{q}\), which results in the following set of possible rational roots:
\[
\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18
\]

Let me know if you'd like to explore part (b) or further clarifications. Here are 5 related questions:

1. How can we test each potential rational root?
2. How do we factor a cubic equation once a root is found?
3. What is the role of synthetic division in solving cubic equations?
4. How does the Rational Zero Theorem help with higher degree polynomials?
5. Can irrational or complex roots exist in this equation?

**Tip:** Always start by testing small values (like \(\pm 1\)) when applying the Rational Zero Theorem to find roots quickly.

List all rational roots that are possible according to the Rational Zero Theorem for the equation x^3 - 2x^2 - 9x + 18 = 0.

Learn how to find all possible rational roots for the polynomial equation x^3 - 2x^2 - 9x + 18 = 0 using the Rational Root Theorem. This guide breaks down the factors of the constant and leading terms to list potential rational roots. Ideal for high school students in Grades 10-12.

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