division of polynomials Math Worksheets, Math Practice for Kids. - Free Printable

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:

\[
\frac{45x^3 + 72x^2 + 9x}{9x}
\]

#### Solution:
1. Divide each term in the numerator by \(9x\):
\[
\frac{45x^3}{9x} = 5x^2, \quad \frac{72x^2}{9x} = 8x, \quad \frac{9x}{9x} = 1
\]
2. Combine the results:
\[
\frac{45x^3 + 72x^2 + 9x}{9x} = 5x^2 + 8x + 1
\]

Answer:
\[
\boxed{5x^2 + 8x + 1}
\]

---

Problem 2:

\[
\frac{40x^2 + 41x - 21}{8x - 3}
\]

#### Solution:
1. Perform polynomial long division:
- Divide the leading term of the numerator (\(40x^2\)) by the leading term of the denominator (\(8x\)):
\[
\frac{40x^2}{8x} = 5x
\]
- Multiply \(5x\) by the entire divisor \(8x - 3\):
\[
5x \cdot (8x - 3) = 40x^2 - 15x
\]
- Subtract this from the original polynomial:
\[
(40x^2 + 41x - 21) - (40x^2 - 15x) = 56x - 21
\]
- Divide the leading term of the new polynomial (\(56x\)) by the leading term of the divisor (\(8x\)):
\[
\frac{56x}{8x} = 7
\]
- Multiply \(7\) by the entire divisor \(8x - 3\):
\[
7 \cdot (8x - 3) = 56x - 21
\]
- Subtract this from the current polynomial:
\[
(56x - 21) - (56x - 21) = 0
\]

2. The quotient is \(5x + 7\) and the remainder is \(0\).

Answer:
\[
\boxed{5x + 7}
\]

---

Problem 3:

\[
\frac{12x^3 + 8x^2 - 40x - 16}{2x + 4}
\]

#### Solution:
1. Perform polynomial long division:
- Divide the leading term of the numerator (\(12x^3\)) by the leading term of the denominator (\(2x\)):
\[
\frac{12x^3}{2x} = 6x^2
\]
- Multiply \(6x^2\) by the entire divisor \(2x + 4\):
\[
6x^2 \cdot (2x + 4) = 12x^3 + 24x^2
\]
- Subtract this from the original polynomial:
\[
(12x^3 + 8x^2 - 40x - 16) - (12x^3 + 24x^2) = -16x^2 - 40x - 16
\]
- Divide the leading term of the new polynomial (\(-16x^2\)) by the leading term of the divisor (\(2x\)):
\[
\frac{-16x^2}{2x} = -8x
\]
- Multiply \(-8x\) by the entire divisor \(2x + 4\):
\[
-8x \cdot (2x + 4) = -16x^2 - 32x
\]
- Subtract this from the current polynomial:
\[
(-16x^2 - 40x - 16) - (-16x^2 - 32x) = -8x - 16
\]
- Divide the leading term of the new polynomial (\(-8x\)) by the leading term of the divisor (\(2x\)):
\[
\frac{-8x}{2x} = -4
\]
- Multiply \(-4\) by the entire divisor \(2x + 4\):
\[
-4 \cdot (2x + 4) = -8x - 16
\]
- Subtract this from the current polynomial:
\[
(-8x - 16) - (-8x - 16) = 0
\]

2. The quotient is \(6x^2 - 8x - 4\) and the remainder is \(0\).

Answer:
\[
\boxed{6x^2 - 8x - 4}
\]

---

Problem 4:

\[
\frac{8x^3 + 14x^2 - 25x - 25}{2x + 5}
\]

#### Solution:
1. Perform polynomial long division:
- Divide the leading term of the numerator (\(8x^3\)) by the leading term of the denominator (\(2x\)):
\[
\frac{8x^3}{2x} = 4x^2
\]
- Multiply \(4x^2\) by the entire divisor \(2x + 5\):
\[
4x^2 \cdot (2x + 5) = 8x^3 + 20x^2
\]
- Subtract this from the original polynomial:
\[
(8x^3 + 14x^2 - 25x - 25) - (8x^3 + 20x^2) = -6x^2 - 25x - 25
\]
- Divide the leading term of the new polynomial (\(-6x^2\)) by the leading term of the divisor (\(2x\)):
\[
\frac{-6x^2}{2x} = -3x
\]
- Multiply \(-3x\) by the entire divisor \(2x + 5\):
\[
-3x \cdot (2x + 5) = -6x^2 - 15x
\]
- Subtract this from the current polynomial:
\[
(-6x^2 - 25x - 25) - (-6x^2 - 15x) = -10x - 25
\]
- Divide the leading term of the new polynomial (\(-10x\)) by the leading term of the divisor (\(2x\)):
\[
\frac{-10x}{2x} = -5
\]
- Multiply \(-5\) by the entire divisor \(2x + 5\):
\[
-5 \cdot (2x + 5) = -10x - 25
\]
- Subtract this from the current polynomial:
\[
(-10x - 25) - (-10x - 25) = 0
\]

2. The quotient is \(4x^2 - 3x - 5\) and the remainder is \(0\).

Answer:
\[
\boxed{4x^2 - 3x - 5}
\]

---

Problem 5:

\[
\frac{6x^3 + 13x^2 - 10x + 15}{x + 3}
\]

#### Solution:
1. Perform polynomial long division:
- Divide the leading term of the numerator (\(6x^3\)) by the leading term of the denominator (\(x\)):
\[
\frac{6x^3}{x} = 6x^2
\]
- Multiply \(6x^2\) by the entire divisor \(x + 3\):
\[
6x^2 \cdot (x + 3) = 6x^3 + 18x^2
\]
- Subtract this from the original polynomial:
\[
(6x^3 + 13x^2 - 10x + 15) - (6x^3 + 18x^2) = -5x^2 - 10x + 15
\]
- Divide the leading term of the new polynomial (\(-5x^2\)) by the leading term of the divisor (\(x\)):
\[
\frac{-5x^2}{x} = -5x
\]
- Multiply \(-5x\) by the entire divisor \(x + 3\):
\[
-5x \cdot (x + 3) = -5x^2 - 15x
\]
- Subtract this from the current polynomial:
\[
(-5x^2 - 10x + 15) - (-5x^2 - 15x) = 5x + 15
\]
- Divide the leading term of the new polynomial (\(5x\)) by the leading term of the divisor (\(x\)):
\[
\frac{5x}{x} = 5
\]
- Multiply \(5\) by the entire divisor \(x + 3\):
\[
5 \cdot (x + 3) = 5x + 15
\]
- Subtract this from the current polynomial:
\[
(5x + 15) - (5x + 15) = 0
\]

2. The quotient is \(6x^2 - 5x + 5\) and the remainder is \(0\).

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

---

Problem 6:

\[
\frac{18x^3 + 21x^2 + 27x}{3x}
\]

#### Solution:
1. Divide each term in the numerator by \(3x\):
\[
\frac{18x^3}{3x} = 6x^2, \quad \frac{21x^2}{3x} = 7x, \quad \frac{27x}{3x} = 9
\]
2. Combine the results:
\[
\frac{18x^3 + 21x^2 + 27x}{3x} = 6x^2 + 7x + 9
\]

Answer:
\[
\boxed{6x^2 + 7x + 9}
\]

---

Problem 7:

\[
\frac{12x^3 - 6x^2 + 16x}{2x}
\]

#### Solution:
1. Divide each term in the numerator by \(2x\):
\[
\frac{12x^3}{2x} = 6x^2, \quad \frac{-6x^2}{2x} = -3x, \quad \frac{16x}{2x} = 8
\]
2. Combine the results:
\[
\frac{12x^3 - 6x^2 + 16x}{2x} = 6x^2 - 3x + 8
\]

Answer:
\[
\boxed{6x^2 - 3x + 8}
\]

---

Problem 8:

\[
\frac{4x^3 - 16x^2 + 12x}{2x}
\]

#### Solution:
1. Divide each term in the numerator by \(2x\):
\[
\frac{4x^3}{2x} = 2x^2, \quad \frac{-16x^2}{2x} = -8x, \quad \frac{12x}{2x} = 6
\]
2. Combine the results:
\[
\frac{4x^3 - 16x^2 + 12x}{2x} = 2x^2 - 8x + 6
\]

Answer:
\[
\boxed{2x^2 - 8x + 6}
\]

---

Problem 9:

\[
\frac{12x^3 + 30x^2 - 30x}{6x}
\]

#### Solution:
1. Divide each term in the numerator by \(6x\):
\[
\frac{12x^3}{6x} = 2x^2, \quad \frac{30x^2}{6x} = 5x, \quad \frac{-30x}{6x} = -5
\]
2. Combine the results:
\[
\frac{12x^3 + 30x^2 - 30x}{6x} = 2x^2 + 5x - 5
\]

Answer:
\[
\boxed{2x^2 + 5x - 5}
\]

---

Problem 10:

\[
\frac{54x^2 + 33x + 3}{9x + 1}
\]

#### Solution:
1. Perform polynomial long division:
- Divide the leading term of the numerator (\(54x^2\)) by the leading term of the denominator (\(9x\)):
\[
\frac{54x^2}{9x} = 6x
\]
- Multiply \(6x\) by the entire divisor \(9x + 1\):
\[
6x \cdot (9x + 1) = 54x^2 + 6x
\]
- Subtract this from the original polynomial:
\[
(54x^2 + 33x + 3) - (54x^2 + 6x) = 27x + 3
\]
- Divide the leading term of the new polynomial (\(27x\)) by the leading term of the divisor (\(9x\)):
\[
\frac{27x}{9x} = 3
\]
- Multiply \(3\) by the entire divisor \(9x + 1\):
\[
3 \cdot (9x + 1) = 27x + 3
\]
- Subtract this from the current polynomial:
\[
(27x + 3) - (27x + 3) = 0
\]

2. The quotient is \(6x + 3\) and the remainder is \(0\).

Answer:
\[
\boxed{6x + 3}
\]

---

Final Answers:

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