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Let's solve this step-by-step using synthetic division to divide \( 3x^4 - 8x^3 - 18x^2 + 8x \) by \( x - 4 \).

### Step 1: Set up synthetic division

1. **Identify the divisor**: \( x - 4 \) means we are dividing by \( 4 \) (since we use the opposite sign of the constant).
   
2. **Write the coefficients of the dividend**:
   - \( 3x^4 \) → Coefficient is 3
   - \( -8x^3 \) → Coefficient is -8
   - \( -18x^2 \) → Coefficient is -18
   - \( +8x \) → Coefficient is 8
   - There is no constant term, so the coefficient for \( x^0 \) is 0.

   So, the coefficients are: \( 3, -8, -18, 8, 0 \).

### Step 2: Perform synthetic division

We will now perform the division using the synthetic division method.

| 4 | 3 | -8 | -18 | 8 | 0 |
|---|---|----|-----|---|---|
|   |   | 12 | 16  | -8| 0 |
|---|---|----|-----|---|---|
| 3 | 4  | -2  | -10 | 0 |

### Explanation:

- Start by bringing down the first coefficient (3) directly.
- Multiply 3 by 4 (the divisor) and write 12 under the next coefficient (-8).
- Add: \( -8 + 12 = 4 \).
- Multiply 4 by 4 and write 16 under the next coefficient (-18).
- Add: \( -18 + 16 = -2 \).
- Multiply -2 by 4 and write -8 under the next coefficient (8).
- Add: \( 8 + (-8) = 0 \).
- Multiply 0 by 4 and write it under the final constant (0).
- Add: \( 0 + 0 = 0 \), so the remainder is 0.

### Final Answer:

The quotient is \( 3x^3 + 4x^2 - 2x - 10 \), and there is no remainder.

Would you like more details on any part of this process? Here are some related questions for practice:

1. What if the divisor was \( x + 4 \) instead of \( x - 4 \)?
2. Can you use long division for the same polynomial division and compare?
3. How would the process change if there was a constant term in the polynomial?
4. What happens in synthetic division when there's a remainder?
5. How does synthetic division relate to factoring polynomials?

**Tip**: Always double-check your synthetic division steps, especially when performing addition and multiplication to avoid small calculation errors!

Use synthetic division to divide 3x^4 - 8x^3 - 18x^2 + 8x by x - 4.

Learn how to perform synthetic division to divide the polynomial 3x^4 - 8x^3 - 18x^2 + 8x by x - 4. This step-by-step solution explains the process of dividing polynomials using synthetic division. Suitable for students in Grades 10-12.

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