https://file.aiquickdraw.com/mathbot/file/1726065325560useyn3r1.jpg

The task requires dividing the polynomials from **Column A** using synthetic division and matching the answers to the options given in **Column B**.

Let's go through each problem step-by-step:

### 1. \( (2x^3 + 3x^2 - 15x - 36) \div (x - 3) \)

Using synthetic division:

- Coefficients of \( 2x^3 + 3x^2 - 15x - 36 \): \( 2, 3, -15, -36 \)
- The divisor \( x - 3 \) means we use \( 3 \) in synthetic division.

Performing the synthetic division:

| 3 | 2 | 3 | -15 | -36 |
|---|---|---|-----|-----|
|   |   | 6 | 27  | 36  |
|---|---|---|-----|-----|
| 2 | 9 | 12 | 0  |     |

Thus, the quotient is \( 2x^2 + 9x + 12 \).

**Answer: (A)**.

### 2. \( (x^3 + 4x^2 - 7x - 10) \div (x - 2) \)

Using synthetic division:

- Coefficients of \( x^3 + 4x^2 - 7x - 10 \): \( 1, 4, -7, -10 \)
- The divisor \( x - 2 \) means we use \( 2 \).

Performing the synthetic division:

| 2 | 1 | 4  | -7  | -10 |
|---|---|----|-----|-----|
|   |    | 2  | 12  | 10  |
|---|---|----|-----|-----|
| 1 | 6  | 5  | 0   |     |

Thus, the quotient is \( x^2 + 6x + 5 \).

**Answer: (B)**.

### 3. \( (2x^3 + 5x^2 - 7x - 12) \div (x + 3) \)

Using synthetic division:

- Coefficients of \( 2x^3 + 5x^2 - 7x - 12 \): \( 2, 5, -7, -12 \)
- The divisor \( x + 3 \) means we use \( -3 \).

Performing the synthetic division:

| -3 | 2 | 5  | -7  | -12 |
|----|---|----|-----|-----|
|    |   | -6 | 3   | 12  |
|----|---|----|-----|-----|
| 2  | -1 | -10 | 0  |     |

Thus, the quotient is \( 2x^2 - x - 10 \).

**Answer: (U)**.

### 4. \( (x^4 - 5x^2 - 10x - 16) \div (x + 2) \)

Using synthetic division:

- Coefficients of \( x^4 - 5x^2 - 10x - 16 \): \( 1, 0, -5, -10, -16 \)
- The divisor \( x + 2 \) means we use \( -2 \).

Performing the synthetic division:

| -2 | 1 | 0 | -5  | -10 | -16 |
|----|---|---|-----|-----|-----|
|    |   | -2| 4   | 2   | -4  |
|----|---|---|-----|-----|-----|
| 1  | -2 | -1 | -6  | 0   |     |

Thus, the quotient is \( x^3 - 2x^2 - x - 6 \).

**Answer: (E)**.

### 5. \( (4x^3 + 2x^2 + 2x - 21) \div (2x - 3) \)

Using synthetic division:

- Coefficients of \( 4x^3 + 2x^2 + 2x - 21 \): \( 4, 2, 2, -21 \)
- The divisor \( 2x - 3 \) means we use \( \frac{3}{2} \).

Performing the synthetic division:

| \( \frac{3}{2} \) | 4  | 2   | 2   | -21 |
|-------------------|----|-----|-----|-----|
|                   |    | 6   | 12  | 21  |
|-------------------|----|-----|-----|-----|
| 4                 | 8  | 8   | 0   |     |

Thus, the quotient is \( 2x^2 + 4x + 7 \).

**Answer: (V)**.

### Final Corresponding Letters:
1. **A**
2. **B**
3. **U**
4. **E**
5. **V**

Would you like any more details or help with these steps?

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Here are 5 follow-up questions to extend your understanding:
1. How do you decide which value to use in synthetic division based on the divisor?
2. What is the remainder when a division results in a quotient with no leftover terms?
3. How does synthetic division simplify polynomial division compared to long division?
4. Can synthetic division be used if the divisor is not in the form \( x - c \)? If so, how?
5. What are the potential uses of synthetic division in real-world applications?

**Tip:** Synthetic division is a great shortcut for dividing polynomials, especially when the divisor is linear (in the form \( x - c \)).

Divide the following polynomials using synthetic division and match the answers to the options given in Column B.

Learn how to divide polynomials using synthetic division. This worksheet provides step-by-step guidance on dividing polynomials and matching answers to given options. Suitable for students in Grades 9-11.

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