Synthetic division (3×³+×²-22×-25) ÷ (×-2)

To solve \((3x^3 + x^2 - 22x - 25) \div (x - 2)\) using synthetic division, follow these steps:

1. **Set up the synthetic division table.**
   - The divisor is \(x - 2\), so we'll use \(2\) in synthetic division.
   - The coefficients of the dividend \(3x^3 + x^2 - 22x - 25\) are: \(3, 1, -22, -25\).

   Set up the synthetic division like this:

   ```
   2 |  3   1   -22   -25
   ```

2. **Bring down the first coefficient (3).**

   ```
   2 |  3   1   -22   -25
       |    
       ---------------
         3
   ```

3. **Multiply 2 by the first number in the bottom row (3), and add it to the next coefficient (1).**
   - \(2 \times 3 = 6\), and \(1 + 6 = 7\).

   ```
   2 |  3   1   -22   -25
       |      6
       ---------------
         3   7
   ```

4. **Multiply 2 by the new number in the bottom row (7), and add it to the next coefficient (-22).**
   - \(2 \times 7 = 14\), and \(-22 + 14 = -8\).

   ```
   2 |  3   1   -22   -25
       |      6    14
       ---------------
         3   7   -8
   ```

5. **Multiply 2 by the new number in the bottom row (-8), and add it to the next coefficient (-25).**
   - \(2 \times -8 = -16\), and \(-25 + (-16) = -41\).

   ```
   2 |  3   1   -22   -25
       |      6    14   -16
       ---------------
         3   7   -8   -41
   ```

6. **Write the quotient and remainder.**
   - The numbers in the bottom row represent the coefficients of the quotient: \(3x^2 + 7x - 8\).
   - The last number (-41) is the remainder.

   So, the result of the division is:
   \[
   3x^2 + 7x - 8 - \frac{41}{x - 2}
   \]

### Final Answer:
\[
3x^2 + 7x - 8 - \frac{41}{x - 2}
\]

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

---

Here are 5 related questions:
1. How does synthetic division compare to long division of polynomials?
2. Can you solve another division problem like \((4x^3 + 5x^2 - 3x + 6) \div (x + 1)\)?
3. What happens when the remainder is zero in synthetic division?
4. How do we factor polynomials using synthetic division?
5. Can synthetic division be used with divisors of degree higher than 1?

**Tip:** Always check that the divisor is of the form \(x - r\) before applying synthetic division.

Synthetic division (3x³ + x² - 22x - 25) ÷ (x - 2)

Learn how to use synthetic division to divide the polynomial 3x³ + x² - 22x - 25 by (x - 2). Follow the step-by-step solution to understand the key concepts of algebra, including synthetic division and the remainder theorem. Ideal for high school students in Grades 10-12.

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