Division of Polynomials by Monomials worksheet with problems and a percentage score grid.

Let’s solve each problem step by step. We are dividing polynomials — some by monomials, some by binomials. I’ll go one at a time and check my work.

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Problem 1: (6x³ + 9x²) ÷ 3x

Divide each term by 3x:

- 6x³ ÷ 3x = 2x²
- 9x² ÷ 3x = 3x

✔ Answer: 2x² + 3x

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Problem 2: (x² - 11x + 18) ÷ (x - 2)

We can factor the numerator or use polynomial division.

Try factoring: Find two numbers that multiply to 18 and add to -11 → -9 and -2.

So: x² - 11x + 18 = (x - 9)(x - 2)

Now divide by (x - 2):

→ (x - 9)(x - 2) ÷ (x - 2) = x - 9

✔ Answer: x - 9

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Problem 3: (8ab + 27a²b² - 3ab²) ÷ ab

Rewrite in order: (27a²b² + 8ab - 3ab²) ÷ ab

Divide each term by ab:

- 27a²b² ÷ ab = 27ab
- 8ab ÷ ab = 8
- -3ab² ÷ ab = -3b

✔ Answer: 27ab + 8 - 3b

*(Note: Order doesn’t matter unless specified; we can write as 27ab - 3b + 8)*

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Problem 4: (y⁶ - 4x³y³) ÷ y³

Divide each term by y³:

- y⁶ ÷ y³ = y³
- -4x³y³ ÷ y³ = -4x³

✔ Answer: y³ - 4x³

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Problem 5: (x² - 15x + 56) ÷ (x - 7)

Factor numerator: Need two numbers that multiply to 56 and add to -15 → -7 and -8.

So: x² - 15x + 56 = (x - 7)(x - 8)

Divide by (x - 7):

→ (x - 7)(x - 8) ÷ (x - 7) = x - 8

✔ Answer: x - 8

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Problem 6: (2x³y² + x²y³) ÷ x²y²

Divide each term:

- 2x³y² ÷ x²y² = 2x
- x²y³ ÷ x²y² = y

✔ Answer: 2x + y

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Problem 7: (4x⁴ - x²) ÷ x²

Divide each term:

- 4x⁴ ÷ x² = 4x²
- -x² ÷ x² = -1

✔ Answer: 4x² - 1

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Problem 8: (10b³ - 4b² - 20b²) ÷ 4b²

Wait — combine like terms first in numerator:

10b³ - 4b² - 20b² = 10b³ - 24b²

Now divide by 4b²:

- 10b³ ÷ 4b² = (10/4)b = (5/2)b
- -24b² ÷ 4b² = -6

✔ Answer: (5/2)b - 6
*(or 2.5b - 6 if decimals allowed, but fraction is better)*

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Problem 9: (5x⁵ - 3x⁴ + 6x + x) ÷ x

First, combine like terms: 6x + x = 7x

So: (5x⁵ - 3x⁴ + 7x) ÷ x

Divide each term by x:

- 5x⁵ ÷ x = 5x⁴
- -3x⁴ ÷ x = -3x³
- 7x ÷ x = 7

✔ Answer: 5x⁴ - 3x³ + 7

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Problem 10: (x² - 6x + 20) ÷ (x - 5)

This does NOT factor nicely. Let’s do polynomial long division.

Divide x² - 6x + 20 by x - 5.

Step 1: x² ÷ x = x → multiply (x - 5) by x → x² - 5x
Subtract: (x² - 6x) - (x² - 5x) = -x
Bring down +20 → now have -x + 20

Step 2: -x ÷ x = -1 → multiply (x - 5) by -1 → -x + 5
Subtract: (-x + 20) - (-x + 5) = 15

So quotient is x - 1, remainder 15.

Answer: x - 1 + 15/(x - 5)

But since this is likely expecting simplified form with remainder, we write:

✔ Answer: x - 1 R 15
*(or as expression: x - 1 + \frac{15}{x - 5})*

However, looking back at other problems, they all divided evenly except maybe this one. Let me double-check the original problem.

Original says: (x² - 6x + 20) ÷ (x - 5)

Check: (x - 5)(x - 1) = x² - x - 5x + 5 = x² - 6x + 5 → not 20.

Difference: 20 - 5 = 15 → so yes, remainder 15.

So final answer must include remainder.

In many school contexts, they accept “quotient with remainder” notation.

✔ Final for #10: x - 1 R 15

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Now let’s list all answers clearly:

1. 2x² + 3x
2. x - 9
3. 27ab + 8 - 3b
4. y³ - 4x³
5. x - 8
6. 2x + y
7. 4x² - 1
8. (5/2)b - 6
9. 5x⁴ - 3x³ + 7
10. x - 1 R 15

I’ve checked each calculation carefully.

Final Answer:
1. 2x² + 3x
2. x - 9
3. 27ab - 3b + 8
4. y³ - 4x³
5. x - 8
6. 2x + y
7. 4x² - 1
8. \frac{5}{2}b - 6
9. 5x⁴ - 3x³ + 7
10. x - 1 \text{ R } 15