Math worksheet titled "Division Of Polynomials" with six problems requiring polynomial division, designed for educational use.
Division of Polynomials worksheet with six polynomial division problems, featuring a header with icons and spaces for name, date, and score.
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Step-by-step solution for: division of polynomials Math Worksheets, Math Practice for Kids.
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Show Answer Key & Explanations
Step-by-step solution for: division of polynomials Math Worksheets, Math Practice for Kids.
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.
---
\[
\frac{45x^3 + 72x^2 + 9x}{9x}
\]
#### Solution:
We can simplify this by dividing each term in the numerator by \(9x\):
\[
\frac{45x^3}{9x} + \frac{72x^2}{9x} + \frac{9x}{9x}
\]
\[
= 5x^2 + 8x + 1
\]
Answer:
\[
\boxed{5x^2 + 8x + 1}
\]
---
\[
\frac{40x^2 + 41x - 21}{8x - 3}
\]
#### Solution:
We perform polynomial long division:
1. Divide the leading term of the numerator by the leading term of the denominator:
\[
\frac{40x^2}{8x} = 5x
\]
2. Multiply \(5x\) by the entire divisor \(8x - 3\):
\[
5x \cdot (8x - 3) = 40x^2 - 15x
\]
3. Subtract this result from the original polynomial:
\[
(40x^2 + 41x - 21) - (40x^2 - 15x) = 56x - 21
\]
4. Repeat the process with the new polynomial \(56x - 21\):
\[
\frac{56x}{8x} = 7
\]
5. Multiply \(7\) by the entire divisor \(8x - 3\):
\[
7 \cdot (8x - 3) = 56x - 21
\]
6. Subtract this result from the current polynomial:
\[
(56x - 21) - (56x - 21) = 0
\]
The quotient is \(5x + 7\) and the remainder is \(0\).
Answer:
\[
\boxed{5x + 7}
\]
---
\[
\frac{12x^3 + 8x^2 - 40x - 16}{2x + 4}
\]
#### Solution:
We perform polynomial long division:
1. Divide the leading term of the numerator by the leading term of the denominator:
\[
\frac{12x^3}{2x} = 6x^2
\]
2. Multiply \(6x^2\) by the entire divisor \(2x + 4\):
\[
6x^2 \cdot (2x + 4) = 12x^3 + 24x^2
\]
3. Subtract this result from the original polynomial:
\[
(12x^3 + 8x^2 - 40x - 16) - (12x^3 + 24x^2) = -16x^2 - 40x - 16
\]
4. Repeat the process with the new polynomial \(-16x^2 - 40x - 16\):
\[
\frac{-16x^2}{2x} = -8x
\]
5. Multiply \(-8x\) by the entire divisor \(2x + 4\):
\[
-8x \cdot (2x + 4) = -16x^2 - 32x
\]
6. Subtract this result from the current polynomial:
\[
(-16x^2 - 40x - 16) - (-16x^2 - 32x) = -8x - 16
\]
7. Repeat the process with the new polynomial \(-8x - 16\):
\[
\frac{-8x}{2x} = -4
\]
8. Multiply \(-4\) by the entire divisor \(2x + 4\):
\[
-4 \cdot (2x + 4) = -8x - 16
\]
9. Subtract this result from the current polynomial:
\[
(-8x - 16) - (-8x - 16) = 0
\]
The quotient is \(6x^2 - 8x - 4\) and the remainder is \(0\).
Answer:
\[
\boxed{6x^2 - 8x - 4}
\]
---
\[
\frac{8x^3 + 14x^2 - 25x - 25}{2x + 5}
\]
#### Solution:
We perform polynomial long division:
1. Divide the leading term of the numerator by the leading term of the denominator:
\[
\frac{8x^3}{2x} = 4x^2
\]
2. Multiply \(4x^2\) by the entire divisor \(2x + 5\):
\[
4x^2 \cdot (2x + 5) = 8x^3 + 20x^2
\]
3. Subtract this result from the original polynomial:
\[
(8x^3 + 14x^2 - 25x - 25) - (8x^3 + 20x^2) = -6x^2 - 25x - 25
\]
4. Repeat the process with the new polynomial \(-6x^2 - 25x - 25\):
\[
\frac{-6x^2}{2x} = -3x
\]
5. Multiply \(-3x\) by the entire divisor \(2x + 5\):
\[
-3x \cdot (2x + 5) = -6x^2 - 15x
\]
6. Subtract this result from the current polynomial:
\[
(-6x^2 - 25x - 25) - (-6x^2 - 15x) = -10x - 25
\]
7. Repeat the process with the new polynomial \(-10x - 25\):
\[
\frac{-10x}{2x} = -5
\]
8. Multiply \(-5\) by the entire divisor \(2x + 5\):
\[
-5 \cdot (2x + 5) = -10x - 25
\]
9. Subtract this result from the current polynomial:
\[
(-10x - 25) - (-10x - 25) = 0
\]
The quotient is \(4x^2 - 3x - 5\) and the remainder is \(0\).
Answer:
\[
\boxed{4x^2 - 3x - 5}
\]
---
\[
\frac{6x^3 + 13x^2 - 10x + 15}{x + 3}
\]
#### Solution:
We perform polynomial long division:
1. Divide the leading term of the numerator by the leading term of the denominator:
\[
\frac{6x^3}{x} = 6x^2
\]
2. Multiply \(6x^2\) by the entire divisor \(x + 3\):
\[
6x^2 \cdot (x + 3) = 6x^3 + 18x^2
\]
3. Subtract this result from the original polynomial:
\[
(6x^3 + 13x^2 - 10x + 15) - (6x^3 + 18x^2) = -5x^2 - 10x + 15
\]
4. Repeat the process with the new polynomial \(-5x^2 - 10x + 15\):
\[
\frac{-5x^2}{x} = -5x
\]
5. Multiply \(-5x\) by the entire divisor \(x + 3\):
\[
-5x \cdot (x + 3) = -5x^2 - 15x
\]
6. Subtract this result from the current polynomial:
\[
(-5x^2 - 10x + 15) - (-5x^2 - 15x) = 5x + 15
\]
7. Repeat the process with the new polynomial \(5x + 15\):
\[
\frac{5x}{x} = 5
\]
8. Multiply \(5\) by the entire divisor \(x + 3\):
\[
5 \cdot (x + 3) = 5x + 15
\]
9. Subtract this result from the current polynomial:
\[
(5x + 15) - (5x + 15) = 0
\]
The quotient is \(6x^2 - 5x + 5\) and the remainder is \(0\).
Answer:
\[
\boxed{6x^2 - 5x + 5}
\]
---
\[
\frac{18x^3 + 21x^2 + 27x}{3x}
\]
#### Solution:
We can simplify this by dividing each term in the numerator by \(3x\):
\[
\frac{18x^3}{3x} + \frac{21x^2}{3x} + \frac{27x}{3x}
\]
\[
= 6x^2 + 7x + 9
\]
Answer:
\[
\boxed{6x^2 + 7x + 9}
\]
---
\[
\frac{12x^3 - 6x^2 + 16x}{2x}
\]
#### Solution:
We can simplify this by dividing each term in the numerator by \(2x\):
\[
\frac{12x^3}{2x} + \frac{-6x^2}{2x} + \frac{16x}{2x}
\]
\[
= 6x^2 - 3x + 8
\]
Answer:
\[
\boxed{6x^2 - 3x + 8}
\]
---
\[
\frac{4x^3 - 16x^2 + 12x}{2x}
\]
#### Solution:
We can simplify this by dividing each term in the numerator by \(2x\):
\[
\frac{4x^3}{2x} + \frac{-16x^2}{2x} + \frac{12x}{2x}
\]
\[
= 2x^2 - 8x + 6
\]
Answer:
\[
\boxed{2x^2 - 8x + 6}
\]
---
\[
\frac{12x^3 + 30x^2 - 30x}{6x}
\]
#### Solution:
We can simplify this by dividing each term in the numerator by \(6x\):
\[
\frac{12x^3}{6x} + \frac{30x^2}{6x} + \frac{-30x}{6x}
\]
\[
= 2x^2 + 5x - 5
\]
Answer:
\[
\boxed{2x^2 + 5x - 5}
\]
---
\[
\frac{54x^2 + 33x + 3}{9x + 1}
\]
#### Solution:
We perform polynomial long division:
1. Divide the leading term of the numerator by the leading term of the denominator:
\[
\frac{54x^2}{9x} = 6x
\]
2. Multiply \(6x\) by the entire divisor \(9x + 1\):
\[
6x \cdot (9x + 1) = 54x^2 + 6x
\]
3. Subtract this result from the original polynomial:
\[
(54x^2 + 33x + 3) - (54x^2 + 6x) = 27x + 3
\]
4. Repeat the process with the new polynomial \(27x + 3\):
\[
\frac{27x}{9x} = 3
\]
5. Multiply \(3\) by the entire divisor \(9x + 1\):
\[
3 \cdot (9x + 1) = 27x + 3
\]
6. Subtract this result from the current polynomial:
\[
(27x + 3) - (27x + 3) = 0
\]
The quotient is \(6x + 3\) and the remainder is \(0\).
Answer:
\[
\boxed{6x + 3}
\]
---
\[
\boxed{
\begin{aligned}
1. & \ 5x^2 + 8x + 1 \\
2. & \ 5x + 7 \\
3. & \ 6x^2 - 8x - 4 \\
4. & \ 4x^2 - 3x - 5 \\
5. & \ 6x^2 - 5x + 5 \\
6. & \ 6x^2 + 7x + 9 \\
7. & \ 6x^2 - 3x + 8 \\
8. & \ 2x^2 - 8x + 6 \\
9. & \ 2x^2 + 5x - 5 \\
10. & \ 6x + 3
\end{aligned}
}
\]
---
Problem 1:
\[
\frac{45x^3 + 72x^2 + 9x}{9x}
\]
#### Solution:
We can simplify this by dividing each term in the numerator by \(9x\):
\[
\frac{45x^3}{9x} + \frac{72x^2}{9x} + \frac{9x}{9x}
\]
\[
= 5x^2 + 8x + 1
\]
Answer:
\[
\boxed{5x^2 + 8x + 1}
\]
---
Problem 2:
\[
\frac{40x^2 + 41x - 21}{8x - 3}
\]
#### Solution:
We perform polynomial long division:
1. Divide the leading term of the numerator by the leading term of the denominator:
\[
\frac{40x^2}{8x} = 5x
\]
2. Multiply \(5x\) by the entire divisor \(8x - 3\):
\[
5x \cdot (8x - 3) = 40x^2 - 15x
\]
3. Subtract this result from the original polynomial:
\[
(40x^2 + 41x - 21) - (40x^2 - 15x) = 56x - 21
\]
4. Repeat the process with the new polynomial \(56x - 21\):
\[
\frac{56x}{8x} = 7
\]
5. Multiply \(7\) by the entire divisor \(8x - 3\):
\[
7 \cdot (8x - 3) = 56x - 21
\]
6. Subtract this result from the current polynomial:
\[
(56x - 21) - (56x - 21) = 0
\]
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:
We perform polynomial long division:
1. Divide the leading term of the numerator by the leading term of the denominator:
\[
\frac{12x^3}{2x} = 6x^2
\]
2. Multiply \(6x^2\) by the entire divisor \(2x + 4\):
\[
6x^2 \cdot (2x + 4) = 12x^3 + 24x^2
\]
3. Subtract this result from the original polynomial:
\[
(12x^3 + 8x^2 - 40x - 16) - (12x^3 + 24x^2) = -16x^2 - 40x - 16
\]
4. Repeat the process with the new polynomial \(-16x^2 - 40x - 16\):
\[
\frac{-16x^2}{2x} = -8x
\]
5. Multiply \(-8x\) by the entire divisor \(2x + 4\):
\[
-8x \cdot (2x + 4) = -16x^2 - 32x
\]
6. Subtract this result from the current polynomial:
\[
(-16x^2 - 40x - 16) - (-16x^2 - 32x) = -8x - 16
\]
7. Repeat the process with the new polynomial \(-8x - 16\):
\[
\frac{-8x}{2x} = -4
\]
8. Multiply \(-4\) by the entire divisor \(2x + 4\):
\[
-4 \cdot (2x + 4) = -8x - 16
\]
9. Subtract this result from the current polynomial:
\[
(-8x - 16) - (-8x - 16) = 0
\]
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:
We perform polynomial long division:
1. Divide the leading term of the numerator by the leading term of the denominator:
\[
\frac{8x^3}{2x} = 4x^2
\]
2. Multiply \(4x^2\) by the entire divisor \(2x + 5\):
\[
4x^2 \cdot (2x + 5) = 8x^3 + 20x^2
\]
3. Subtract this result from the original polynomial:
\[
(8x^3 + 14x^2 - 25x - 25) - (8x^3 + 20x^2) = -6x^2 - 25x - 25
\]
4. Repeat the process with the new polynomial \(-6x^2 - 25x - 25\):
\[
\frac{-6x^2}{2x} = -3x
\]
5. Multiply \(-3x\) by the entire divisor \(2x + 5\):
\[
-3x \cdot (2x + 5) = -6x^2 - 15x
\]
6. Subtract this result from the current polynomial:
\[
(-6x^2 - 25x - 25) - (-6x^2 - 15x) = -10x - 25
\]
7. Repeat the process with the new polynomial \(-10x - 25\):
\[
\frac{-10x}{2x} = -5
\]
8. Multiply \(-5\) by the entire divisor \(2x + 5\):
\[
-5 \cdot (2x + 5) = -10x - 25
\]
9. Subtract this result from the current polynomial:
\[
(-10x - 25) - (-10x - 25) = 0
\]
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:
We perform polynomial long division:
1. Divide the leading term of the numerator by the leading term of the denominator:
\[
\frac{6x^3}{x} = 6x^2
\]
2. Multiply \(6x^2\) by the entire divisor \(x + 3\):
\[
6x^2 \cdot (x + 3) = 6x^3 + 18x^2
\]
3. Subtract this result from the original polynomial:
\[
(6x^3 + 13x^2 - 10x + 15) - (6x^3 + 18x^2) = -5x^2 - 10x + 15
\]
4. Repeat the process with the new polynomial \(-5x^2 - 10x + 15\):
\[
\frac{-5x^2}{x} = -5x
\]
5. Multiply \(-5x\) by the entire divisor \(x + 3\):
\[
-5x \cdot (x + 3) = -5x^2 - 15x
\]
6. Subtract this result from the current polynomial:
\[
(-5x^2 - 10x + 15) - (-5x^2 - 15x) = 5x + 15
\]
7. Repeat the process with the new polynomial \(5x + 15\):
\[
\frac{5x}{x} = 5
\]
8. Multiply \(5\) by the entire divisor \(x + 3\):
\[
5 \cdot (x + 3) = 5x + 15
\]
9. Subtract this result from the current polynomial:
\[
(5x + 15) - (5x + 15) = 0
\]
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:
We can simplify this by dividing each term in the numerator by \(3x\):
\[
\frac{18x^3}{3x} + \frac{21x^2}{3x} + \frac{27x}{3x}
\]
\[
= 6x^2 + 7x + 9
\]
Answer:
\[
\boxed{6x^2 + 7x + 9}
\]
---
Problem 7:
\[
\frac{12x^3 - 6x^2 + 16x}{2x}
\]
#### Solution:
We can simplify this by dividing each term in the numerator by \(2x\):
\[
\frac{12x^3}{2x} + \frac{-6x^2}{2x} + \frac{16x}{2x}
\]
\[
= 6x^2 - 3x + 8
\]
Answer:
\[
\boxed{6x^2 - 3x + 8}
\]
---
Problem 8:
\[
\frac{4x^3 - 16x^2 + 12x}{2x}
\]
#### Solution:
We can simplify this by dividing each term in the numerator by \(2x\):
\[
\frac{4x^3}{2x} + \frac{-16x^2}{2x} + \frac{12x}{2x}
\]
\[
= 2x^2 - 8x + 6
\]
Answer:
\[
\boxed{2x^2 - 8x + 6}
\]
---
Problem 9:
\[
\frac{12x^3 + 30x^2 - 30x}{6x}
\]
#### Solution:
We can simplify this by dividing each term in the numerator by \(6x\):
\[
\frac{12x^3}{6x} + \frac{30x^2}{6x} + \frac{-30x}{6x}
\]
\[
= 2x^2 + 5x - 5
\]
Answer:
\[
\boxed{2x^2 + 5x - 5}
\]
---
Problem 10:
\[
\frac{54x^2 + 33x + 3}{9x + 1}
\]
#### Solution:
We perform polynomial long division:
1. Divide the leading term of the numerator by the leading term of the denominator:
\[
\frac{54x^2}{9x} = 6x
\]
2. Multiply \(6x\) by the entire divisor \(9x + 1\):
\[
6x \cdot (9x + 1) = 54x^2 + 6x
\]
3. Subtract this result from the original polynomial:
\[
(54x^2 + 33x + 3) - (54x^2 + 6x) = 27x + 3
\]
4. Repeat the process with the new polynomial \(27x + 3\):
\[
\frac{27x}{9x} = 3
\]
5. Multiply \(3\) by the entire divisor \(9x + 1\):
\[
3 \cdot (9x + 1) = 27x + 3
\]
6. Subtract this result from the current polynomial:
\[
(27x + 3) - (27x + 3) = 0
\]
The quotient is \(6x + 3\) and the remainder is \(0\).
Answer:
\[
\boxed{6x + 3}
\]
---
Final Answers:
\[
\boxed{
\begin{aligned}
1. & \ 5x^2 + 8x + 1 \\
2. & \ 5x + 7 \\
3. & \ 6x^2 - 8x - 4 \\
4. & \ 4x^2 - 3x - 5 \\
5. & \ 6x^2 - 5x + 5 \\
6. & \ 6x^2 + 7x + 9 \\
7. & \ 6x^2 - 3x + 8 \\
8. & \ 2x^2 - 8x + 6 \\
9. & \ 2x^2 + 5x - 5 \\
10. & \ 6x + 3
\end{aligned}
}
\]
Parent Tip: Review the logic above to help your child master the concept of long division polynomials worksheet.
