To determine all possible rational zeros of the function $f(x) = x^3 + 9x^2 - 213x - 221$ using the Rational Zero Theorem, follow these steps:

Rational Zero Theorem Explanation:

The potential rational zeros of a polynomial are of the form: $\pm \frac{\text{factors of the constant term (} a_0 \text{)}}{\text{factors of the leading coefficient (} a_n \text{)}}$

1. Identify coefficients:

• Constant term ($a_0$): $-221$
• Leading coefficient ($a_n$): $1$

2. Find factors of $a_0$ (constant term):

The factors of $-221$ are: $\pm 1, \pm 13, \pm 17, \pm 221$

3. Find factors of $a_n$ (leading coefficient):

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

4. Generate all possible rational zeros:

Divide each factor of $a_0$ by each factor of $a_n$. Since $a_n = 1$, the possible rational zeros are: $\pm 1, \pm 13, \pm 17, \pm 221$

Final Answer:

The possible rational zeros are: $\boxed{\pm 1, \pm 13, \pm 17, \pm 221}$

Would you like to evaluate which of these are actual zeros of the polynomial? Let me know if you want a detailed explanation or further assistance!

Here are 5 related questions:

1. What are the steps to confirm if a given rational zero is an actual zero?
2. How does synthetic division help in verifying potential rational zeros?
3. Can irrational or complex roots also exist for this polynomial? Why?
4. How do we find all real zeros of this polynomial completely?
5. What role does the degree of the polynomial play in determining the number of zeros?

Tip: When using the Rational Zero Theorem, always simplify the factors of $a_0$ and $a_n$ to avoid missing any possible zeros.