Match the polynomial division problems on the left with their quoitents on the right.

Column A

1.

(12x3 + 27x2 + 75x + 65) / (3x + 3)

:

(12x3 + 27x2 + 75x + 65) / (3x + 3)

2.

(15x3 + 37x2 + 53x + 55) / (3x + 5)

:

(15x3 + 37x2 + 53x + 55) / (3x + 5)

3.

(16x3 + 44x2 - 40x + 7) / (2x + 7)

:

(16x3 + 44x2 - 40x + 7) / (2x + 7)

4.

(x3 + 7x2 + 11x + 2) / (x + 2)

:

(x3 + 7x2 + 11x + 2) / (x + 2)

Column B

a.

-3x2 + 5x + 2

b.

5x2 + 4x + 11

c.

8x2 -6x + 1

d.

4x2 - 3x + 20 remainder 6

e.

4x2 + 5x + 20 remainder 5

f.

x2 + 5x + 1

To match the polynomial division problems with their quotients, let's perform each division step-by-step:

1. **(12x³ + 27x² + 75x + 65) / (3x + 3)**:
   - **Quotient**: \(4x² + 5x + 20\), remainder \(5\) (matches with **e**).

2. **(15x³ + 37x² + 53x + 55) / (3x + 5)**:
   - **Quotient**: \(5x² + 4x + 11\) (matches with **b**).

3. **(16x³ + 44x² - 40x + 7) / (2x + 7)**:
   - **Quotient**: \(8x² - 6x + 1\) (matches with **c**).

4. **(x³ + 7x² + 11x + 2) / (x + 2)**:
   - **Quotient**: \(x² + 5x + 1\) (matches with **f**).

So the matches are:
- 1: e
- 2: b
- 3: c
- 4: f

If you need more details or have any questions, let me know! 

Here are some follow-up questions you might consider:
1. Do you need help with the long division steps for any of these problems?
2. Would you like to see examples of synthetic division?
3. Are you interested in learning about the remainder theorem?
4. Would you like to explore more complex polynomial divisions?
5. Do you have additional polynomial problems you want to solve? 

**Tip**: Always check your work by multiplying the quotient by the divisor and adding the remainder; it should equal the original polynomial.

Match the polynomial division problems on the left with their quotients on the right.

Explore how to match polynomial division problems with their correct quotients. This guide provides detailed solutions for polynomial divisions, including steps and explanations, suitable for high school students.

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