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To divide \( x^3 + x^2 - 4x + 3 \) by \( x - 2 \) using synthetic division, we follow these steps:

1. Write down the coefficients of \( x^3 + x^2 - 4x + 3 \): \( 1, 1, -4, 3 \).
2. Since we're dividing by \( x - 2 \), the number we use in synthetic division is \( 2 \).

Let's proceed with the synthetic division:

### Synthetic Division Steps:

1. **Write down the coefficients**: \( 1, 1, -4, 3 \).
2. **Bring down** the first coefficient \( 1 \) as it is.
3. **Multiply** this \( 1 \) by \( 2 \) and write the result (\( 2 \)) below the next coefficient:
   - The second column becomes \( 1 + 2 = 3 \).
4. **Multiply** \( 3 \) by \( 2 \) and write the result (\( 6 \)) below the next coefficient:
   - The third column becomes \( -4 + 6 = 2 \).
5. **Multiply** \( 2 \) by \( 2 \) and write the result (\( 4 \)) below the last coefficient:
   - The fourth column becomes \( 3 + 4 = 7 \).

### Result:

The bottom row represents the coefficients of the quotient and the remainder:
- **Quotient**: \( x^2 + 3x + 2 \)
- **Remainder**: \( 7 \)

So, the answer is:
- **Quotient**: \( x^2 + 3x + 2 \)
- **Remainder**: \( 7 \)

Would you like further details on this process or have any questions?

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### Related Questions:

1. How do you apply synthetic division to higher-degree polynomials?
2. What happens if the divisor is in a different form, such as \( x + a \)?
3. How can synthetic division help find factors of a polynomial?
4. How is synthetic division different from long division of polynomials?
5. How does the remainder theorem relate to synthetic division?

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### Tip:
When using synthetic division, always ensure to bring down the first coefficient directly and multiply consistently to avoid mistakes.

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

Learn how to divide the polynomial x^3 + x^2 - 4x + 3 by x - 2 using synthetic division. This problem covers key algebraic concepts, including synthetic division and the remainder theorem. Suitable for high school students in Grades 9-12.

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