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To solve the problem using synthetic division, let's break down the polynomial division of:

\[
\frac{x^4 - x^3 + x^2 - x + 2}{x - 3}
\]

### Step-by-Step Synthetic Division

1. **Set up the synthetic division:**  
   Since we are dividing by \( x - 3 \), the value we will use for synthetic division is \( 3 \).

2. **Write the coefficients of the polynomial:**  
   The coefficients of \( x^4 - x^3 + x^2 - x + 2 \) are:  
   \[
   [1, -1, 1, -1, 2]
   \]

3. **Perform the synthetic division:**

|        | 3  |      |      |      |      |
|--------|----|------|------|------|------|
| 1      |    |      |      |      |      |
|        |    | 3    | 6    | 21   | 60   |
| 1      | -1 | 1    | -1   | 2    |      |
|        | 1  | 2    | 7    | 20   | 62   |

- Bring down the first coefficient (1).
- Multiply \( 1 \times 3 = 3 \), and add to the next coefficient \( -1 + 3 = 2 \).
- Multiply \( 2 \times 3 = 6 \), and add to the next coefficient \( 1 + 6 = 7 \).
- Multiply \( 7 \times 3 = 21 \), and add to the next coefficient \( -1 + 21 = 20 \).
- Multiply \( 20 \times 3 = 60 \), and add to the next constant term \( 2 + 60 = 62 \).

### Results:
- The quotient is \( x^3 + 2x^2 + 7x + 20 \).
- The remainder is \( 62 \).

Thus, the result of the synthetic division is:

\[
\text{Quotient: } x^3 + 2x^2 + 7x + 20
\]
\[
\text{Remainder: } 62
\]

Would you like a more detailed explanation of synthetic division or have any questions?

Here are some related questions to consider:

1. How can you use synthetic division to verify the remainder theorem?
2. What happens if the divisor is \( x + 3 \) instead of \( x - 3 \)?
3. How does synthetic division differ from long division of polynomials?
4. Can synthetic division be used for divisors of the form \( x^2 - a \)?
5. What role does the remainder play in evaluating polynomials at certain values?

**Tip:** Synthetic division is most efficient when the divisor is of the form \( x - a \). For more complex divisors, long division may be required.

Find the quotient and remainder using synthetic division.

Learn how to perform synthetic division with the polynomial x^4 - x^3 + x^2 - x + 2 divided by x - 3. Discover the steps of bringing down, multiplying, and adding coefficients sequentially to find the quotient and remainder. Suitable for high school students studying algebra.

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