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To solve this problem, we will use synthetic division to divide the polynomial \( f(x) = x^3 - 15x^2 + 68x - 96 \) by \( x - 3 \).

### Step 1: Set up Synthetic Division
Since we are dividing by \( x - 3 \), we place \( 3 \) to the left. The coefficients of the polynomial are \( 1, -15, 68, \) and \( -96 \).

```
  3 |  1   -15   68   -96
      |      (3) (-36)   96
      -------------------------
        1   -12   32     0
```

### Step 2: Perform Synthetic Division
1. Bring down the first coefficient \( 1 \).
2. Multiply \( 1 \) by \( 3 \) and add to the next coefficient:
   - \( -15 + 3 = -12 \)
3. Multiply \( -12 \) by \( 3 \) and add to the next coefficient:
   - \( 68 + (-36) = 32 \)
4. Multiply \( 32 \) by \( 3 \) and add to the next coefficient:
   - \( -96 + 96 = 0 \)

The remainder is \( 0 \), so the division is exact. The quotient is \( x^2 - 12x + 32 \).

### Step 3: Factor the Quotient
Now, we factor \( x^2 - 12x + 32 = 0 \).
1. The factors of \( 32 \) that add up to \( -12 \) are \( -4 \) and \( -8 \).
2. So, \( x^2 - 12x + 32 = (x - 4)(x - 8) \).

### Step 4: Find All Zeros
The zeros of \( f(x) \) are the solutions to \( x - 3 = 0 \), \( x - 4 = 0 \), and \( x - 8 = 0 \). Therefore:
\[
x = 3, 4, 8
\]

### Answer
- **Quotient**: \( x^2 - 12x + 32 \)
- **Zeros** of \( f \): \( 3, 4, 8 \)

Would you like further details or have questions on any part of the solution?

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### Additional Related Questions
1. How do you use synthetic division to check if a root is valid for a polynomial?
2. What are the steps to factor a quadratic polynomial?
3. How do you determine the quotient and remainder in synthetic division?
4. Why does the remainder being zero confirm that \( x - 3 \) is a factor of \( f(x) \)?
5. Can synthetic division be used for polynomials with complex coefficients?

### Tip
When using synthetic division, always verify each multiplication and addition step to avoid small mistakes in large polynomials.

Use synthetic division to divide f(x) = x^3 - 15x^2 + 68x - 96 by x - 3. Use the result to find all zeros of f.

Learn how to use synthetic division to divide the polynomial f(x) = x^3 - 15x^2 + 68x - 96 by x - 3 and find all zeros of f. This solution covers polynomial division, synthetic division steps, and factoring techniques. Suitable for high school students.

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