Use synthetic division to find the quotient and the remainder.
left parenthesis 3 x cubed minus 3 x squared plus x minus 8 right parenthesisdivided byleft parenthesis x plus 1 right parenthesis

To use synthetic division to divide \(3x^3 - 3x^2 + x - 8\) by \(x + 1\), follow these steps:

1. **Set up the synthetic division**: Since we are dividing by \(x + 1\), use the opposite sign of \(+1\), which is \(-1\).
   
2. **Write down the coefficients** of the polynomial \(3x^3 - 3x^2 + x - 8\): 
   - Coefficients are \(3\), \(-3\), \(1\), and \(-8\).

3. **Synthetic Division Process**:
   - Bring down the first coefficient, \(3\), as it is.
   - Multiply \(3\) by \(-1\) (the divisor) and add to the next coefficient.
   - Repeat this process across all coefficients.

Let's proceed step-by-step:

|       |  -1    |     |       |
|-------|--------|-----|-------|
| Coefficients | 3     | -3  | 1    | -8  |
| Steps       | ↓     |     |      |
| Step 1: Bring down \(3\)      | 3     |     |      |
| Multiply \(3 \times -1 = -3\) |       | -3  |      |
| Add: \(-3 + (-3) = -6\)       | 3     | -6  |      |
| Multiply \(-6 \times -1 = 6\) |       |     | 6    |
| Add: \(1 + 6 = 7\)            | 3     | -6  | 7    |
| Multiply \(7 \times -1 = -7\) |       |     |      |
| Add: \(-8 + (-7) = -15\)      | 3     | -6  | 7    | -15 |

4. **Interpret the result**:
   - The quotient coefficients are \(3x^2 - 6x + 7\).
   - The remainder is \(-15\).

### Final Answer:
The result of dividing \(3x^3 - 3x^2 + x - 8\) by \(x + 1\) is:
\[
3x^2 - 6x + 7 \text{ with a remainder of } -15.
\]
So, we can write:
\[
\frac{3x^3 - 3x^2 + x - 8}{x + 1} = 3x^2 - 6x + 7 - \frac{15}{x + 1}.
\]

Would you like more details on synthetic division, or do you have any questions?

### Further Questions:
1. How would you handle synthetic division with a divisor like \(x - 2\) instead of \(x + 1\)?
2. What is the purpose of synthetic division in polynomial division?
3. Can you perform synthetic division if the leading coefficient is not 1? How would you adjust?
4. How does synthetic division differ from long division of polynomials?
5. How can we verify the result of synthetic division?

**Tip**: Remember, synthetic division only works when dividing by linear factors of the form \(x - c\).

Use synthetic division to find the quotient and the remainder. (3x^3 - 3x^2 + x - 8) divided by (x + 1)

Learn how to perform synthetic division to divide the polynomial 3x^3 - 3x^2 + x - 8 by x + 1. This step-by-step guide explains the process and shows how to obtain the quotient and remainder. Suitable for students in Grades 9-11 studying polynomial division.

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