Use synthetic division to find the result when 3, x, cubed, minus, 3, x, squared, plus, 2, x, minus, 163x 
3
 −3x 
2
 +2x−16 is divided by x, minus, 2x−2.

To solve this problem using synthetic division, let's go through the steps methodically. 

### Step 1: Identify the polynomial and the divisor
The given polynomial is:
\[
3x^3 - 3x^2 + 2x - 16
\]

The divisor is:
\[
x - 2
\]

### Step 2: Set up the synthetic division
We will use the root of the divisor \( x - 2 \) (which is \( x = 2 \)) for synthetic division. 

### Step 3: List the coefficients of the polynomial
The coefficients of the polynomial \( 3x^3 - 3x^2 + 2x - 16 \) are:

\[
3, -3, 2, -16
\]

### Step 4: Perform synthetic division
We will perform synthetic division with the divisor \( x = 2 \).

1. **Write down the coefficients**: \( 3, -3, 2, -16 \)
2. **Bring down the leading coefficient**: Bring down the \( 3 \) as it is.
3. **Multiply and add**:
   - Multiply \( 3 \) by \( 2 \), and add to the next coefficient: \( 3 \times 2 = 6 \), \( -3 + 6 = 3 \).
   - Multiply \( 3 \) by \( 2 \), and add to the next coefficient: \( 3 \times 2 = 6 \), \( 2 + 6 = 8 \).
   - Multiply \( 8 \) by \( 2 \), and add to the next coefficient: \( 8 \times 2 = 16 \), \( -16 + 16 = 0 \).

The final row gives the coefficients of the quotient and the remainder.

\[
\begin{array}{r|rrrr}
2 & 3 & -3 & 2 & -16 \\
  &   & 6  & 6 & 16  \\
\hline
  & 3 & 3  & 8 & 0   \\
\end{array}
\]

### Step 5: Interpret the result
The synthetic division yields:
\[
3x^2 + 3x + 8 \text{ with a remainder of } 0
\]

Thus, the quotient is \( 3x^2 + 3x + 8 \) and the remainder is \( 0 \).

### Final Answer
The result of the division is \( 3x^2 + 3x + 8 \).

---

Do you want further details or have any questions? Here are five related questions:

1. How is synthetic division different from long division?
2. What are some other applications of synthetic division?
3. Can synthetic division be used for dividing by polynomials with higher degrees?
4. What happens if there is a non-zero remainder in synthetic division?
5. How would the process differ if the divisor was \( x + 2 \) instead of \( x - 2 \)?

**Tip:** Always double-check the coefficients and signs when performing synthetic division to avoid errors.

Learn how to use synthetic division to divide the polynomial 3x^3 - 3x^2 + 2x - 16 by x - 2 step-by-step. Understand the process, coefficients, and interpretation of the result. Suitable for high school students studying polynomial division.

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