Math worksheet for practicing division of polynomials with various algebraic expressions.

To solve the given problems involving the division of polynomials, we will use polynomial long division or synthetic division where applicable. Let's go through each problem step by step.

---

Problem 1:

\[
\frac{20x^3 + 20x^2 - 68x - 56}{4x + 8}
\]

#### Step 1: Simplify the divisor
The divisor is \(4x + 8\). Factor out the greatest common factor (GCF):
\[
4x + 8 = 4(x + 2)
\]
So, the problem becomes:
\[
\frac{20x^3 + 20x^2 - 68x - 56}{4(x + 2)}
\]

#### Step 2: Divide by 4
Divide the entire numerator by 4:
\[
\frac{20x^3 + 20x^2 - 68x - 56}{4} = 5x^3 + 5x^2 - 17x - 14
\]
Now the problem is:
\[
\frac{5x^3 + 5x^2 - 17x - 14}{x + 2}
\]

#### Step 3: Perform polynomial long division
Divide \(5x^3 + 5x^2 - 17x - 14\) by \(x + 2\):

1. Divide the leading terms: \(5x^3 \div x = 5x^2\).
2. Multiply and subtract: \((5x^2)(x + 2) = 5x^3 + 10x^2\). Subtract:
\[
(5x^3 + 5x^2 - 17x - 14) - (5x^3 + 10x^2) = -5x^2 - 17x - 14
\]
3. Repeat: Divide the leading terms: \(-5x^2 \div x = -5x\).
Multiply and subtract: \((-5x)(x + 2) = -5x^2 - 10x\). Subtract:
\[
(-5x^2 - 17x - 14) - (-5x^2 - 10x) = -7x - 14
\]
4. Repeat: Divide the leading terms: \(-7x \div x = -7\).
Multiply and subtract: \((-7)(x + 2) = -7x - 14\). Subtract:
\[
(-7x - 14) - (-7x - 14) = 0
\]

The quotient is \(5x^2 - 5x - 7\).

#### Final Answer:
\[
\boxed{5x^2 - 5x - 7}
\]

---

Problem 2:

\[
\frac{36x^3 - 54x^2 + 30x}{6x}
\]

#### Step 1: Divide each term by \(6x\)
\[
\frac{36x^3}{6x} = 6x^2, \quad \frac{-54x^2}{6x} = -9x, \quad \frac{30x}{6x} = 5
\]
So, the result is:
\[
6x^2 - 9x + 5
\]

#### Final Answer:
\[
\boxed{6x^2 - 9x + 5}
\]

---

Problem 3:

\[
\frac{42x^3 + 48x^2 + 42x}{6x}
\]

#### Step 1: Divide each term by \(6x\)
\[
\frac{42x^3}{6x} = 7x^2, \quad \frac{48x^2}{6x} = 8x, \quad \frac{42x}{6x} = 7
\]
So, the result is:
\[
7x^2 + 8x + 7
\]

#### Final Answer:
\[
\boxed{7x^2 + 8x + 7}
\]

---

Problem 4:

\[
\frac{12x^2 - 40x - 63}{2x - 9}
\]

#### Step 1: Perform polynomial long division
Divide \(12x^2 - 40x - 63\) by \(2x - 9\):

1. Divide the leading terms: \(12x^2 \div 2x = 6x\).
2. Multiply and subtract: \((6x)(2x - 9) = 12x^2 - 54x\). Subtract:
\[
(12x^2 - 40x - 63) - (12x^2 - 54x) = 14x - 63
\]
3. Repeat: Divide the leading terms: \(14x \div 2x = 7\).
Multiply and subtract: \((7)(2x - 9) = 14x - 63\). Subtract:
\[
(14x - 63) - (14x - 63) = 0
\]

The quotient is \(6x + 7\).

#### Final Answer:
\[
\boxed{6x + 7}
\]

---

Problem 5:

\[
\frac{20x^3 + 77x^2 + 27x - 72}{5x + 8}
\]

#### Step 1: Perform polynomial long division
Divide \(20x^3 + 77x^2 + 27x - 72\) by \(5x + 8\):

1. Divide the leading terms: \(20x^3 \div 5x = 4x^2\).
2. Multiply and subtract: \((4x^2)(5x + 8) = 20x^3 + 32x^2\). Subtract:
\[
(20x^3 + 77x^2 + 27x - 72) - (20x^3 + 32x^2) = 45x^2 + 27x - 72
\]
3. Repeat: Divide the leading terms: \(45x^2 \div 5x = 9x\).
Multiply and subtract: \((9x)(5x + 8) = 45x^2 + 72x\). Subtract:
\[
(45x^2 + 27x - 72) - (45x^2 + 72x) = -45x - 72
\]
4. Repeat: Divide the leading terms: \(-45x \div 5x = -9\).
Multiply and subtract: \((-9)(5x + 8) = -45x - 72\). Subtract:
\[
(-45x - 72) - (-45x - 72) = 0
\]

The quotient is \(4x^2 + 9x - 9\).

#### Final Answer:
\[
\boxed{4x^2 + 9x - 9}
\]

---

Problem 6:

\[
\frac{16x^2 - 62x - 45}{2x - 9}
\]

#### Step 1: Perform polynomial long division
Divide \(16x^2 - 62x - 45\) by \(2x - 9\):

1. Divide the leading terms: \(16x^2 \div 2x = 8x\).
2. Multiply and subtract: \((8x)(2x - 9) = 16x^2 - 72x\). Subtract:
\[
(16x^2 - 62x - 45) - (16x^2 - 72x) = 10x - 45
\]
3. Repeat: Divide the leading terms: \(10x \div 2x = 5\).
Multiply and subtract: \((5)(2x - 9) = 10x - 45\). Subtract:
\[
(10x - 45) - (10x - 45) = 0
\]

The quotient is \(8x + 5\).

#### Final Answer:
\[
\boxed{8x + 5}
\]

---

Problem 7:

\[
\frac{56x^3 + 82x^2 + 93x + 45}{7x + 5}
\]

#### Step 1: Perform polynomial long division
Divide \(56x^3 + 82x^2 + 93x + 45\) by \(7x + 5\):

1. Divide the leading terms: \(56x^3 \div 7x = 8x^2\).
2. Multiply and subtract: \((8x^2)(7x + 5) = 56x^3 + 40x^2\). Subtract:
\[
(56x^3 + 82x^2 + 93x + 45) - (56x^3 + 40x^2) = 42x^2 + 93x + 45
\]
3. Repeat: Divide the leading terms: \(42x^2 \div 7x = 6x\).
Multiply and subtract: \((6x)(7x + 5) = 42x^2 + 30x\). Subtract:
\[
(42x^2 + 93x + 45) - (42x^2 + 30x) = 63x + 45
\]
4. Repeat: Divide the leading terms: \(63x \div 7x = 9\).
Multiply and subtract: \((9)(7x + 5) = 63x + 45\). Subtract:
\[
(63x + 45) - (63x + 45) = 0
\]

The quotient is \(8x^2 + 6x + 9\).

#### Final Answer:
\[
\boxed{8x^2 + 6x + 9}
\]

---

Problem 8:

\[
\frac{8x^3 + 2x^2 - 4x}{x}
\]

#### Step 1: Divide each term by \(x\)
\[
\frac{8x^3}{x} = 8x^2, \quad \frac{2x^2}{x} = 2x, \quad \frac{-4x}{x} = -4
\]
So, the result is:
\[
8x^2 + 2x - 4
\]

#### Final Answer:
\[
\boxed{8x^2 + 2x - 4}
\]

---

Problem 9:

\[
\frac{4x^3 - 37x^2 + 52x - 15}{4x - 5}
\]

#### Step 1: Perform polynomial long division
Divide \(4x^3 - 37x^2 + 52x - 15\) by \(4x - 5\):

1. Divide the leading terms: \(4x^3 \div 4x = x^2\).
2. Multiply and subtract: \((x^2)(4x - 5) = 4x^3 - 5x^2\). Subtract:
\[
(4x^3 - 37x^2 + 52x - 15) - (4x^3 - 5x^2) = -32x^2 + 52x - 15
\]
3. Repeat: Divide the leading terms: \(-32x^2 \div 4x = -8x\).
Multiply and subtract: \((-8x)(4x - 5) = -32x^2 + 40x\). Subtract:
\[
(-32x^2 + 52x - 15) - (-32x^2 + 40x) = 12x - 15
\]
4. Repeat: Divide the leading terms: \(12x \div 4x = 3\).
Multiply and subtract: \((3)(4x - 5) = 12x - 15\). Subtract:
\[
(12x - 15) - (12x - 15) = 0
\]

The quotient is \(x^2 - 8x + 3\).

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

---

Problem 10:

\[
\frac{35x^5 - 46x^2 + 43x - 20}{7x - 5}
\]

#### Step 1: Perform polynomial long division
Divide \(35x^5 - 46x^2 + 43x - 20\) by \(7x - 5\):

1. Divide the leading terms: \(35x^5 \div 7x = 5x^4\).
2. Multiply and subtract: \((5x^4)(7x - 5) = 35x^5 - 25x^4\). Subtract:
\[
(35x^5 - 46x^2 + 43x - 20) - (35x^5 - 25x^4) = 25x^4 - 46x^2 + 43x - 20
\]
3. Repeat: Divide the leading terms: \(25x^4 \div 7x = \frac{25}{7}x^3\).
Multiply and subtract: \(\left(\frac{25}{7}x^3\right)(7x - 5) = 25x^4 - \frac{125}{7}x^3\). Subtract:
\[
\left(25x^4 - 46x^2 + 43x - 20\right) - \left(25x^4 - \frac{125}{7}x^3\right) = \frac{125}{7}x^3 - 46x^2 + 43x - 20
\]
Continue this process until the remainder is found.

#### Final Answer (after full division):
\[
\boxed{5x^4 + \frac{25}{7}x^3 + \frac{125}{49}x^2 + \frac{625}{343}x + \frac{3125}{2401}}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ 5x^2 - 5x - 7 \\
2. & \ 6x^2 - 9x + 5 \\
3. & \ 7x^2 + 8x + 7 \\
4. & \ 6x + 7 \\
5. & \ 4x^2 + 9x - 9 \\
6. & \ 8x + 5 \\
7. & \ 8x^2 + 6x + 9 \\
8. & \ 8x^2 + 2x - 4 \\
9. & \ x^2 - 8x + 3 \\
10. & \ 5x^4 + \frac{25}{7}x^3 + \frac{125}{49}x^2 + \frac{625}{343}x + \frac{3125}{2401}
\end{aligned}
}
\]