Use the Rational Zero Test to list the possible rational zeros of f. Verify that the zeros of f shown on the graph are contained in the list. (Enter your answers as a comma-separated list.)
f(x) = x cubed − 9x squared − 9x + 81

To list the possible rational zeros of the polynomial function \( f(x) = x^3 - 9x^2 - 9x + 81 \) using the Rational Zero Test, we follow these steps:

### Rational Zero Test:
The possible rational zeros of a polynomial are given by the following formula:

\[
\text{Possible Rational Zeros} = \frac{\text{Factors of the constant term}}{\text{Factors of the leading coefficient}}
\]

1. **Constant term:** The constant term in \( f(x) \) is \( 81 \).
2. **Leading coefficient:** The leading coefficient of \( f(x) \) is the coefficient of \( x^3 \), which is \( 1 \).

#### Step 1: List the factors of the constant term \( 81 \).
The factors of \( 81 \) are:

\[
\pm 1, \pm 3, \pm 9, \pm 27, \pm 81
\]

#### Step 2: List the factors of the leading coefficient \( 1 \).
The factors of \( 1 \) are:

\[
\pm 1
\]

#### Step 3: Use the Rational Zero Test formula.
Now, using the Rational Zero Test, the possible rational zeros are all the factors of \( 81 \) divided by the factors of \( 1 \), which are:

\[
\pm 1, \pm 3, \pm 9, \pm 27, \pm 81
\]

Thus, the possible rational zeros are:

\[
\boxed{\pm 1, \pm 3, \pm 9, \pm 27, \pm 81}
\]

### Verifying the Zeros from the Graph:
If you have a graph of the function \( f(x) = x^3 - 9x^2 - 9x + 81 \), check if any of these values (e.g., \( 1, 3, 9, 27, 81 \)) are zeros of the function, meaning where the graph crosses or touches the x-axis. If any of the zeros match, they are confirmed to be part of the possible rational zeros list.

Would you like help with graphing or further exploration on how to find the actual zeros?

### Related Questions:
1. How do we use synthetic division to confirm the zeros?
2. What happens when a zero is repeated (multiplicity)?
3. Can you explain the difference between rational and irrational zeros?
4. How does the Rational Zero Test relate to the Factor Theorem?
5. How do we simplify factoring cubics?

### Tip:
The Rational Zero Test provides possible rational zeros, but you'll need to test each one or use synthetic division to find the actual zeros.

Use the Rational Zero Test to list the possible rational zeros of f. Verify that the zeros of f shown on the graph are contained in the list. (Enter your answers as a comma-separated list.) f(x) = x cubed − 9x squared − 9x + 81

Learn how to apply the Rational Zero Test to find the possible rational zeros of the polynomial f(x) = x^3 − 9x^2 − 9x + 81. This problem involves key algebraic concepts and provides a step-by-step solution for high school students in Grades 10-12.

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