divide using synthetic division: x³+7x-3x²+5 ÷ x-2

Let's perform synthetic division of \( x^3 + 7x - 3x^2 + 5 \) by \( x - 2 \).

### Step-by-Step Synthetic Division Process:

1. **Arrange the polynomial in standard form:**  
   Rearrange \( x^3 + 7x - 3x^2 + 5 \) to \( x^3 - 3x^2 + 7x + 5 \).

2. **Set up the synthetic division:**  
   We are dividing by \( x - 2 \), so we use \( 2 \) as the divisor.

3. **Write the coefficients of the dividend polynomial:**
   - Coefficients of \( x^3 - 3x^2 + 7x + 5 \) are: \( 1, -3, 7, 5 \).

4. **Perform synthetic division:**

   | Coefficients       | 1    | -3    | 7     | 5    |
   |--------------------|------|-------|-------|------|
   | Divisor \( (x - 2) \) | 2    |       |       |      |
   | Synthetic Process   |      | 2     | -2    | 10   |
   | Result              | 1    | -1    | 5     | 15   |

   **Explanation of the process:**
   - Bring down the leading coefficient (1).
   - Multiply by 2 (the root of \( x - 2 \)): \( 1 \times 2 = 2 \).
   - Add to the next coefficient: \( -3 + 2 = -1 \).
   - Multiply by 2 again: \( -1 \times 2 = -2 \).
   - Add to the next coefficient: \( 7 + (-2) = 5 \).
   - Multiply by 2: \( 5 \times 2 = 10 \).
   - Add to the last coefficient: \( 5 + 10 = 15 \).

### Conclusion:
The quotient is \( x^2 - x + 5 \) and the remainder is \( 15 \).

Thus, the result of the division is:
\[
x^3 - 3x^2 + 7x + 5 \div (x - 2) = x^2 - x + 5 + \frac{15}{x - 2}.
\]

Would you like further details or clarifications?  

Here are 5 related questions:
1. What is synthetic division used for in algebra?
2. How do you interpret a remainder in synthetic division?
3. What happens when you divide by a factor where the divisor is more complex?
4. Can synthetic division be used for divisors that are not linear?
5. How do you verify the result of synthetic division?

**Tip:** Always check the divisor to make sure the equation is set up correctly for synthetic division.

Master synthetic division by solving x³ + 7x - 3x² + 5 ÷ x - 2 with this detailed step-by-step guide. This problem focuses on key algebraic concepts like polynomial division and synthetic division, making it ideal for high school students in Grades 9-12.

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