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Math 10
Copy, read and understand the process of Dividing Polynomials using SYNTHETIC DIVISION. Then answer the same given in Long Method but this time use the Synthetic Division.

The images you uploaded outline examples of synthetic division and a task asking you to apply it to solve polynomial division problems. The examples demonstrate how to perform synthetic division step-by-step and provide the following two problems to solve:

### Given Problems:
1. \( \frac{2x^3 + 9x^2 + 3x - 4}{x + 4} \)
2. \( \frac{8x^3 + 27}{2x + 3} \)

Let’s walk through the synthetic division of each.

### Problem 1: \( \frac{2x^3 + 9x^2 + 3x - 4}{x + 4} \)
1. **Rewrite the divisor**: For synthetic division, you use the zero of the divisor. In this case, the divisor is \(x + 4\), so we use \(a = -4\).
   
2. **Write down the coefficients**: The coefficients of the dividend \(2x^3 + 9x^2 + 3x - 4\) are \( [2, 9, 3, -4] \).

3. **Perform synthetic division**:
   - First, bring down the first coefficient \(2\).
   - Multiply \(2\) by \(-4\) and write the result under the next coefficient.
   - Add the result to the next coefficient and repeat the process.

|  -4   |  2   |  9   |  3   |  -4  |
|-------|------|------|------|------|
|       |      | -8   | -4   | 4    |
|       |  2   |  1   |  -1  |  0   |

   The quotient is \(2x^2 + x - 1\), and the remainder is \(0\).

   So, the division is:
   \[
   \frac{2x^3 + 9x^2 + 3x - 4}{x + 4} = 2x^2 + x - 1
   \]

### Problem 2: \( \frac{8x^3 + 27}{2x + 3} \)
1. **Rewrite the divisor**: The divisor is \(2x + 3\), so we use \(a = -\frac{3}{2}\).

2. **Write down the coefficients**: The coefficients of the dividend \(8x^3 + 27\) are \( [8, 0, 0, 27] \) (since there are no \(x^2\) and \(x\) terms).

3. **Perform synthetic division**:
   - First, bring down the first coefficient \(8\).
   - Multiply \(8\) by \(-\frac{3}{2}\), and proceed with synthetic division.

|  -3/2 |  8   |  0   |  0   |  27  |
|-------|------|------|------|------|
|       |      | -12  |  18  | -27  |
|       |  8   | -12  |  18  |  0   |

   The quotient is \(8x^2 - 12x + 18\), and the remainder is \(0\).

   So, the division is:
   \[
   \frac{8x^3 + 27}{2x + 3} = 8x^2 - 12x + 18
   \]

Would you like further details on any of these steps, or help with additional questions?

### Related Questions:
1. What is the significance of the remainder when performing synthetic division?
2. How do you handle cases where the divisor is not a binomial?
3. Can synthetic division be used for non-integer coefficients in the divisor?
4. How does synthetic division differ from long division in polynomials?
5. What happens if the remainder is not zero?

### Tip:
When the divisor is of the form \(x - c\), synthetic division is much faster than long division. Always ensure to change the sign of the constant when setting up the synthetic division.

Copy, read and understand the process of Dividing Polynomials using SYNTHETIC DIVISION. Then answer the same given in Long Method but this time use the Synthetic Division.

Learn how to divide polynomials using synthetic division. Examples include dividing cubic and quartic polynomials by binomials, illustrating key algebraic steps. Suitable for students in Grades 10-12.

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