This worksheet provides ten practice problems for dividing polynomials, ranging from simple monomial division to binomial long division.

The task involves dividing polynomials. Let's solve each problem step by step.

---

Problem 1:

\[
(24x^4 + 24x^3 + 3x^2) \div 8x^2
\]

#### Solution:
1. Divide each term in the numerator by \(8x^2\):
- \(\frac{24x^4}{8x^2} = 3x^2\)
- \(\frac{24x^3}{8x^2} = 3x\)
- \(\frac{3x^2}{8x^2} = \frac{3}{8}\)

2. Combine the results:
\[
3x^2 + 3x + \frac{3}{8}
\]

Answer:
\[
\boxed{3x^2 + 3x + \frac{3}{8}}
\]

---

Problem 2:

\[
(16x^5 + 2x^4 + x^3) \div 4x
\]

#### Solution:
1. Divide each term in the numerator by \(4x\):
- \(\frac{16x^5}{4x} = 4x^4\)
- \(\frac{2x^4}{4x} = \frac{1}{2}x^3\)
- \(\frac{x^3}{4x} = \frac{1}{4}x^2\)

2. Combine the results:
\[
4x^4 + \frac{1}{2}x^3 + \frac{1}{4}x^2
\]

Answer:
\[
\boxed{4x^4 + \frac{1}{2}x^3 + \frac{1}{4}x^2}
\]

---

Problem 3:

\[
(27x^3 + 9x^2 + 9x) \div 9x
\]

#### Solution:
1. Divide each term in the numerator by \(9x\):
- \(\frac{27x^3}{9x} = 3x^2\)
- \(\frac{9x^2}{9x} = x\)
- \(\frac{9x}{9x} = 1\)

2. Combine the results:
\[
3x^2 + x + 1
\]

Answer:
\[
\boxed{3x^2 + x + 1}
\]

---

Problem 4:

\[
(2x^3 + 8x^2 + 12x) \div 4x^2
\]

#### Solution:
1. Divide each term in the numerator by \(4x^2\):
- \(\frac{2x^3}{4x^2} = \frac{1}{2}x\)
- \(\frac{8x^2}{4x^2} = 2\)
- \(\frac{12x}{4x^2} = \frac{3}{x}\)

2. Combine the results:
\[
\frac{1}{2}x + 2 + \frac{3}{x}
\]

Answer:
\[
\boxed{\frac{1}{2}x + 2 + \frac{3}{x}}
\]

---

Problem 5:

\[
(x^2 + 9x + 17) \div (x + 4)
\]

#### Solution:
Perform polynomial long division:

1. Divide the leading term of the numerator (\(x^2\)) by the leading term of the denominator (\(x\)):
\[
\frac{x^2}{x} = x
\]
Write \(x\) above the division bar.

2. Multiply \(x\) by \(x + 4\):
\[
x \cdot (x + 4) = x^2 + 4x
\]
Subtract this from the original polynomial:
\[
(x^2 + 9x + 17) - (x^2 + 4x) = 5x + 17
\]

3. Divide the leading term of the new polynomial (\(5x\)) by the leading term of the denominator (\(x\)):
\[
\frac{5x}{x} = 5
\]
Write \(5\) above the division bar.

4. Multiply \(5\) by \(x + 4\):
\[
5 \cdot (x + 4) = 5x + 20
\]
Subtract this from the current polynomial:
\[
(5x + 17) - (5x + 20) = -3
\]

The quotient is \(x + 5\) and the remainder is \(-3\). Thus:
\[
x + 5 - \frac{3}{x + 4}
\]

Answer:
\[
\boxed{x + 5 - \frac{3}{x + 4}}
\]

---

Problem 6:

\[
(x^2 - x - 16) \div (x - 4)
\]

#### Solution:
Perform polynomial long division:

1. Divide the leading term of the numerator (\(x^2\)) by the leading term of the denominator (\(x\)):
\[
\frac{x^2}{x} = x
\]
Write \(x\) above the division bar.

2. Multiply \(x\) by \(x - 4\):
\[
x \cdot (x - 4) = x^2 - 4x
\]
Subtract this from the original polynomial:
\[
(x^2 - x - 16) - (x^2 - 4x) = 3x - 16
\]

3. Divide the leading term of the new polynomial (\(3x\)) by the leading term of the denominator (\(x\)):
\[
\frac{3x}{x} = 3
\]
Write \(3\) above the division bar.

4. Multiply \(3\) by \(x - 4\):
\[
3 \cdot (x - 4) = 3x - 12
\]
Subtract this from the current polynomial:
\[
(3x - 16) - (3x - 12) = -4
\]

The quotient is \(x + 3\) and the remainder is \(-4\). Thus:
\[
x + 3 - \frac{4}{x - 4}
\]

Answer:
\[
\boxed{x + 3 - \frac{4}{x - 4}}
\]

---

Problem 7:

\[
(x^2 - 6x + 2) \div (x - 1)
\]

#### Solution:
Perform polynomial long division:

1. Divide the leading term of the numerator (\(x^2\)) by the leading term of the denominator (\(x\)):
\[
\frac{x^2}{x} = x
\]
Write \(x\) above the division bar.

2. Multiply \(x\) by \(x - 1\):
\[
x \cdot (x - 1) = x^2 - x
\]
Subtract this from the original polynomial:
\[
(x^2 - 6x + 2) - (x^2 - x) = -5x + 2
\]

3. Divide the leading term of the new polynomial (\(-5x\)) by the leading term of the denominator (\(x\)):
\[
\frac{-5x}{x} = -5
\]
Write \(-5\) above the division bar.

4. Multiply \(-5\) by \(x - 1\):
\[
-5 \cdot (x - 1) = -5x + 5
\]
Subtract this from the current polynomial:
\[
(-5x + 2) - (-5x + 5) = -3
\]

The quotient is \(x - 5\) and the remainder is \(-3\). Thus:
\[
x - 5 - \frac{3}{x - 1}
\]

Answer:
\[
\boxed{x - 5 - \frac{3}{x - 1}}
\]

---

Problem 8:

\[
(x^2 - x - 24) \div (x - 5)
\]

#### Solution:
Perform polynomial long division:

1. Divide the leading term of the numerator (\(x^2\)) by the leading term of the denominator (\(x\)):
\[
\frac{x^2}{x} = x
\]
Write \(x\) above the division bar.

2. Multiply \(x\) by \(x - 5\):
\[
x \cdot (x - 5) = x^2 - 5x
\]
Subtract this from the original polynomial:
\[
(x^2 - x - 24) - (x^2 - 5x) = 4x - 24
\]

3. Divide the leading term of the new polynomial (\(4x\)) by the leading term of the denominator (\(x\)):
\[
\frac{4x}{x} = 4
\]
Write \(4\) above the division bar.

4. Multiply \(4\) by \(x - 5\):
\[
4 \cdot (x - 5) = 4x - 20
\]
Subtract this from the current polynomial:
\[
(4x - 24) - (4x - 20) = -4
\]

The quotient is \(x + 4\) and the remainder is \(-4\). Thus:
\[
x + 4 - \frac{4}{x - 5}
\]

Answer:
\[
\boxed{x + 4 - \frac{4}{x - 5}}
\]

---

Problem 9:

\[
(2x^2 - 8x + 2) \div (x - 3)
\]

#### Solution:
Perform polynomial long division:

1. Divide the leading term of the numerator (\(2x^2\)) by the leading term of the denominator (\(x\)):
\[
\frac{2x^2}{x} = 2x
\]
Write \(2x\) above the division bar.

2. Multiply \(2x\) by \(x - 3\):
\[
2x \cdot (x - 3) = 2x^2 - 6x
\]
Subtract this from the original polynomial:
\[
(2x^2 - 8x + 2) - (2x^2 - 6x) = -2x + 2
\]

3. Divide the leading term of the new polynomial (\(-2x\)) by the leading term of the denominator (\(x\)):
\[
\frac{-2x}{x} = -2
\]
Write \(-2\) above the division bar.

4. Multiply \(-2\) by \(x - 3\):
\[
-2 \cdot (x - 3) = -2x + 6
\]
Subtract this from the current polynomial:
\[
(-2x + 2) - (-2x + 6) = -4
\]

The quotient is \(2x - 2\) and the remainder is \(-4\). Thus:
\[
2x - 2 - \frac{4}{x - 3}
\]

Answer:
\[
\boxed{2x - 2 - \frac{4}{x - 3}}
\]

---

Problem 10:

\[
(2x^2 + 5x - 11) \div (x - 4)
\]

#### Solution:
Perform polynomial long division:

1. Divide the leading term of the numerator (\(2x^2\)) by the leading term of the denominator (\(x\)):
\[
\frac{2x^2}{x} = 2x
\]
Write \(2x\) above the division bar.

2. Multiply \(2x\) by \(x - 4\):
\[
2x \cdot (x - 4) = 2x^2 - 8x
\]
Subtract this from the original polynomial:
\[
(2x^2 + 5x - 11) - (2x^2 - 8x) = 13x - 11
\]

3. Divide the leading term of the new polynomial (\(13x\)) by the leading term of the denominator (\(x\)):
\[
\frac{13x}{x} = 13
\]
Write \(13\) above the division bar.

4. Multiply \(13\) by \(x - 4\):
\[
13 \cdot (x - 4) = 13x - 52
\]
Subtract this from the current polynomial:
\[
(13x - 11) - (13x - 52) = 41
\]

The quotient is \(2x + 13\) and the remainder is \(41\). Thus:
\[
2x + 13 + \frac{41}{x - 4}
\]

Answer:
\[
\boxed{2x + 13 + \frac{41}{x - 4}}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \quad 3x^2 + 3x + \frac{3}{8} \\
2. & \quad 4x^4 + \frac{1}{2}x^3 + \frac{1}{4}x^2 \\
3. & \quad 3x^2 + x + 1 \\
4. & \quad \frac{1}{2}x + 2 + \frac{3}{x} \\
5. & \quad x + 5 - \frac{3}{x + 4} \\
6. & \quad x + 3 - \frac{4}{x - 4} \\
7. & \quad x - 5 - \frac{3}{x - 1} \\
8. & \quad x + 4 - \frac{4}{x - 5} \\
9. & \quad 2x - 2 - \frac{4}{x - 3} \\
10. & \quad 2x + 13 + \frac{41}{x - 4}
\end{aligned}
}
\]