Math worksheet for practicing polynomial division by monomials.

Problem: Dividing Polynomials by a Monomial (Single Term)

The task involves dividing each polynomial by the given monomial. The general approach is to divide each term of the polynomial by the monomial and simplify the result.

Let's solve each problem step by step.

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1. \( (6x^3 + 30x^2 + 24x) \div 12 \)

#### Step 1: Divide each term by 12.
- \( \frac{6x^3}{12} = \frac{6}{12} \cdot x^3 = \frac{1}{2}x^3 \)
- \( \frac{30x^2}{12} = \frac{30}{12} \cdot x^2 = \frac{5}{2}x^2 \)
- \( \frac{24x}{12} = \frac{24}{12} \cdot x = 2x \)

#### Final Answer:
\[ \frac{1}{2}x^3 + \frac{5}{2}x^2 + 2x \]

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2. \( (15x^4 + 20x^3 + 5x) \div 5 \)

#### Step 1: Divide each term by 5.
- \( \frac{15x^4}{5} = \frac{15}{5} \cdot x^4 = 3x^4 \)
- \( \frac{20x^3}{5} = \frac{20}{5} \cdot x^3 = 4x^3 \)
- \( \frac{5x}{5} = \frac{5}{5} \cdot x = x \)

#### Final Answer:
\[ 3x^4 + 4x^3 + x \]

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3. \( (14x^3 + 10x^2 + 6x) \div 6x \)

#### Step 1: Divide each term by \( 6x \).
- \( \frac{14x^3}{6x} = \frac{14}{6} \cdot \frac{x^3}{x} = \frac{7}{3}x^2 \)
- \( \frac{10x^2}{6x} = \frac{10}{6} \cdot \frac{x^2}{x} = \frac{5}{3}x \)
- \( \frac{6x}{6x} = \frac{6}{6} \cdot \frac{x}{x} = 1 \)

#### Final Answer:
\[ \frac{7}{3}x^2 + \frac{5}{3}x + 1 \]

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4. \( (17x^2 + 81x^{17} + 9x) \div 9x \)

#### Step 1: Divide each term by \( 9x \).
- \( \frac{17x^2}{9x} = \frac{17}{9} \cdot \frac{x^2}{x} = \frac{17}{9}x \)
- \( \frac{81x^{17}}{9x} = \frac{81}{9} \cdot \frac{x^{17}}{x} = 9x^{16} \)
- \( \frac{9x}{9x} = \frac{9}{9} \cdot \frac{x}{x} = 1 \)

#### Final Answer:
\[ 9x^{16} + \frac{17}{9}x + 1 \]

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5. \( (60x^{17} + 60x^{10} + 30x^9) \div 30x \)

#### Step 1: Divide each term by \( 30x \).
- \( \frac{60x^{17}}{30x} = \frac{60}{30} \cdot \frac{x^{17}}{x} = 2x^{16} \)
- \( \frac{60x^{10}}{30x} = \frac{60}{30} \cdot \frac{x^{10}}{x} = 2x^9 \)
- \( \frac{30x^9}{30x} = \frac{30}{30} \cdot \frac{x^9}{x} = x^8 \)

#### Final Answer:
\[ 2x^{16} + 2x^9 + x^8 \]

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6. \( (60x^{17} + 24x^{10} + 24x^7) \div 12x^7 \)

#### Step 1: Divide each term by \( 12x^7 \).
- \( \frac{60x^{17}}{12x^7} = \frac{60}{12} \cdot \frac{x^{17}}{x^7} = 5x^{10} \)
- \( \frac{24x^{10}}{12x^7} = \frac{24}{12} \cdot \frac{x^{10}}{x^7} = 2x^3 \)
- \( \frac{24x^7}{12x^7} = \frac{24}{12} \cdot \frac{x^7}{x^7} = 2 \)

#### Final Answer:
\[ 5x^{10} + 2x^3 + 2 \]

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7. \( (33x^7 + 99x^5 + 11x^4) \div 33x \)

#### Step 1: Divide each term by \( 33x \).
- \( \frac{33x^7}{33x} = \frac{33}{33} \cdot \frac{x^7}{x} = x^6 \)
- \( \frac{99x^5}{33x} = \frac{99}{33} \cdot \frac{x^5}{x} = 3x^4 \)
- \( \frac{11x^4}{33x} = \frac{11}{33} \cdot \frac{x^4}{x} = \frac{1}{3}x^3 \)

#### Final Answer:
\[ x^6 + 3x^4 + \frac{1}{3}x^3 \]

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8. \( (18x^{12} + 21x^7 + 24x^5) \div 6x^2 \)

#### Step 1: Divide each term by \( 6x^2 \).
- \( \frac{18x^{12}}{6x^2} = \frac{18}{6} \cdot \frac{x^{12}}{x^2} = 3x^{10} \)
- \( \frac{21x^7}{6x^2} = \frac{21}{6} \cdot \frac{x^7}{x^2} = \frac{7}{2}x^5 \)
- \( \frac{24x^5}{6x^2} = \frac{24}{6} \cdot \frac{x^5}{x^2} = 4x^3 \)

#### Final Answer:
\[ 3x^{10} + \frac{7}{2}x^5 + 4x^3 \]

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9. \( (28x^{12} - 42x^7 - 28x^5) \div 21x^2 \)

#### Step 1: Divide each term by \( 21x^2 \).
- \( \frac{28x^{12}}{21x^2} = \frac{28}{21} \cdot \frac{x^{12}}{x^2} = \frac{4}{3}x^{10} \)
- \( \frac{-42x^7}{21x^2} = \frac{-42}{21} \cdot \frac{x^7}{x^2} = -2x^5 \)
- \( \frac{-28x^5}{21x^2} = \frac{-28}{21} \cdot \frac{x^5}{x^2} = -\frac{4}{3}x^3 \)

#### Final Answer:
\[ \frac{4}{3}x^{10} - 2x^5 - \frac{4}{3}x^3 \]

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10. \( (-24x^9 + 42x^7 + 42x^5) \div 18x \)

#### Step 1: Divide each term by \( 18x \).
- \( \frac{-24x^9}{18x} = \frac{-24}{18} \cdot \frac{x^9}{x} = -\frac{4}{3}x^8 \)
- \( \frac{42x^7}{18x} = \frac{42}{18} \cdot \frac{x^7}{x} = \frac{7}{3}x^6 \)
- \( \frac{42x^5}{18x} = \frac{42}{18} \cdot \frac{x^5}{x} = \frac{7}{3}x^4 \)

#### Final Answer:
\[ -\frac{4}{3}x^8 + \frac{7}{3}x^6 + \frac{7}{3}x^4 \]

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11. \( (-40x^{12} + 30x^{10} - 10x^7 + 30x^5 + 80x^3) \div 20x^2 \)

#### Step 1: Divide each term by \( 20x^2 \).
- \( \frac{-40x^{12}}{20x^2} = \frac{-40}{20} \cdot \frac{x^{12}}{x^2} = -2x^{10} \)
- \( \frac{30x^{10}}{20x^2} = \frac{30}{20} \cdot \frac{x^{10}}{x^2} = \frac{3}{2}x^8 \)
- \( \frac{-10x^7}{20x^2} = \frac{-10}{20} \cdot \frac{x^7}{x^2} = -\frac{1}{2}x^5 \)
- \( \frac{30x^5}{20x^2} = \frac{30}{20} \cdot \frac{x^5}{x^2} = \frac{3}{2}x^3 \)
- \( \frac{80x^3}{20x^2} = \frac{80}{20} \cdot \frac{x^3}{x^2} = 4x \)

#### Final Answer:
\[ -2x^{10} + \frac{3}{2}x^8 - \frac{1}{2}x^5 + \frac{3}{2}x^3 + 4x \]

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12. \( (77x^{14} + 55x^{13} + 99x^{12} + 88x^{11} + 44x^{10} + 11x^9) \div 11x^9 \)

#### Step 1: Divide each term by \( 11x^9 \).
- \( \frac{77x^{14}}{11x^9} = \frac{77}{11} \cdot \frac{x^{14}}{x^9} = 7x^5 \)
- \( \frac{55x^{13}}{11x^9} = \frac{55}{11} \cdot \frac{x^{13}}{x^9} = 5x^4 \)
- \( \frac{99x^{12}}{11x^9} = \frac{99}{11} \cdot \frac{x^{12}}{x^9} = 9x^3 \)
- \( \frac{88x^{11}}{11x^9} = \frac{88}{11} \cdot \frac{x^{11}}{x^9} = 8x^2 \)
- \( \frac{44x^{10}}{11x^9} = \frac{44}{11} \cdot \frac{x^{10}}{x^9} = 4x \)
- \( \frac{11x^9}{11x^9} = \frac{11}{11} \cdot \frac{x^9}{x^9} = 1 \)

#### Final Answer:
\[ 7x^5 + 5x^4 + 9x^3 + 8x^2 + 4x + 1 \]

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13. \( (-40x^{12} - 20x^{11} - 25x^9 - 30x^8) \div 15x^8 \)

#### Step 1: Divide each term by \( 15x^8 \).
- \( \frac{-40x^{12}}{15x^8} = \frac{-40}{15} \cdot \frac{x^{12}}{x^8} = -\frac{8}{3}x^4 \)
- \( \frac{-20x^{11}}{15x^8} = \frac{-20}{15} \cdot \frac{x^{11}}{x^8} = -\frac{4}{3}x^3 \)
- \( \frac{-25x^9}{15x^8} = \frac{-25}{15} \cdot \frac{x^9}{x^8} = -\frac{5}{3}x \)
- \( \frac{-30x^8}{15x^8} = \frac{-30}{15} \cdot \frac{x^8}{x^8} = -2 \)

#### Final Answer:
\[ -\frac{8}{3}x^4 - \frac{4}{3}x^3 - \frac{5}{3}x - 2 \]

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Final Answers:

1. \( \frac{1}{2}x^3 + \frac{5}{2}x^2 + 2x \)
2. \( 3x^4 + 4x^3 + x \)
3. \( \frac{7}{3}x^2 + \frac{5}{3}x + 1 \)
4. \( 9x^{16} + \frac{17}{9}x + 1 \)
5. \( 2x^{16} + 2x^9 + x^8 \)
6. \( 5x^{10} + 2x^3 + 2 \)
7. \( x^6 + 3x^4 + \frac{1}{3}x^3 \)
8. \( 3x^{10} + \frac{7}{2}x^5 + 4x^3 \)
9. \( \frac{4}{3}x^{10} - 2x^5 - \frac{4}{3}x^3 \)
10. \( -\frac{4}{3}x^8 + \frac{7}{3}x^6 + \frac{7}{3}x^4 \)
11. \( -2x^{10} + \frac{3}{2}x^8 - \frac{1}{2}x^5 + \frac{3}{2}x^3 + 4x \)
12. \( 7x^5 + 5x^4 + 9x^3 + 8x^2 + 4x + 1 \)
13. \( -\frac{8}{3}x^4 - \frac{4}{3}x^3 - \frac{5}{3}x - 2 \)

\boxed{\text{See above for detailed solutions and final answers.}}