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The given problem asks us to simplify the expression \(\frac{x^3 - 9x}{x - 3}\) using synthetic division.

### Step-by-Step Solution:

1. **Set up the synthetic division:**
   - The divisor is \(x - 3\), so we use \(3\) in synthetic division.
   - The coefficients of the polynomial \(x^3 - 9x\) are \( [1, 0, -9, 0] \) (note that there is no \(x^2\) term, so the coefficient is \(0\)).

2. **Perform synthetic division:**
   ```
   3 |  1    0   -9    0
         |       3     9    0
         -----------------
           1    3     0    0
   ```

   Here, the steps are:
   - Bring down the first coefficient, \(1\).
   - Multiply \(1\) by \(3\) (the divisor root), giving \(3\).
   - Add \(0 + 3 = 3\).
   - Multiply \(3\) by \(3\), giving \(9\).
   - Add \(-9 + 9 = 0\).
   - Multiply \(0\) by \(3\), giving \(0\).
   - Add \(0 + 0 = 0\).

3. **Interpret the result:**
   - The quotient is \(x^2 + 3x\).
   - The remainder is \(0\).

4. **Final Answer:**
   Since there is no remainder, the expression simplifies to:
   \[
   x^2 + 3x
   \]

Thus, \(\frac{x^3 - 9x}{x - 3} = x^2 + 3x\).

Would you like any further details or explanations?

### Follow-up Questions:
1. How do you handle synthetic division when there is a remainder?
2. What happens if the divisor has a degree higher than one (e.g., \(x^2 - 3x + 2\))?
3. Can synthetic division be used for dividing by any polynomial?
4. How would the answer change if the polynomial in the numerator had an extra constant term?
5. What does the remainder tell you about the relationship between the divisor and the original polynomial?

### Tip:
In synthetic division, always ensure to include coefficients for missing powers of \(x\), like adding a \(0\) for \(x^2\) when it's absent.

Use synthetic division to simplify (x^3 - 9x) / (x - 3).

Learn how to simplify the expression (x^3 - 9x) divided by (x - 3) using synthetic division. This step-by-step guide breaks down the process, explains key algebraic concepts, and provides the final simplified result. Suitable for students in Grades 9-12.

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