I'll solve each polynomial division problem step by step. When dividing a polynomial by a monomial, I divide each term of the polynomial by the monomial separately.
Let me work through all 13 problems:
Problem 1: (6x³ + 30x² + 24x) ÷ 12
- 6x³ ÷ 12 = x³/2 or 0.5x³
- 30x² ÷ 12 = 5x²/2 or 2.5x²
- 24x ÷ 12 = 2x
Answer: x³/2 + 5x²/2 + 2x or 0.5x³ + 2.5x² + 2x
Problem 2: (15x⁵ + 20x² + 5x) ÷ 5
- 15x⁵ ÷ 5 = 3x⁵
- 20x² ÷ 5 = 4x²
- 5x ÷ 5 = x
Answer: 3x⁵ + 4x² + x
Problem 3: (14x⁴ + 10x² + 6x) ÷ 6x
- 14x⁴ ÷ 6x = 7x³/3
- 10x² ÷ 6x = 5x/3
- 6x ÷ 6x = 1
Answer: 7x³/3 + 5x/3 + 1
Problem 4: (72x⁴ + 81x² + 9x) ÷ 9x
- 72x⁴ ÷ 9x = 8x³
- 81x² ÷ 9x = 9x
- 9x ÷ 9x = 1
Answer: 8x³ + 9x + 1
Problem 5: (60x¹² + 60x¹¹ + 30x⁸) ÷ 30x
- 60x¹² ÷ 30x = 2x¹¹
- 60x¹¹ 30x = 2x¹
- 30x⁸ ÷ 30x = x⁷
Answer: 2x¹¹ + 2x¹ + x⁷
Problem 6: (60x¹² + 24x¹⁰ + 24x⁵) ÷ 12x²
- 60x¹² ÷ 12x² = 5x¹⁰
- 24x¹⁰ ÷ 12x² = 2x⁸
- 24x ÷ 12x² = 2x³
Answer: 5x¹⁰ + 2x⁸ + 2x³
Problem 7: (33x + 99x³ + 11x²) ÷ 33x
- 33x⁵ ÷ 33x = x⁴
- 99x³ ÷ 33x = 3x²
- 11x² ÷ 33x = x/3
Answer: x⁴ + 3x² + x/3
Problem 8: (18x¹⁰ + 21x⁹ + 24x) ÷ 6x²
- 18x¹⁰ ÷ 6x² = 3x⁸
- 21x ÷ 6x² = 7x/2 or 3.5x⁷
- 24x⁷ ÷ 6x² = 4x⁵
Answer: 3x⁸ + 7x⁷/2 + 4x⁵ or 3x⁸ + 3.5x⁷ + 4x⁵
Problem 9: (28x¹² - 42x - 28x⁶) ÷ 21x²
- 28x¹² ÷ 21x² = 4x¹⁰/3
- -42x⁷ ÷ 21x² = -2x⁵
- -28x⁶ ÷ 21x² = -4x⁴/3
Answer: 4x¹⁰/3 - 2x - 4x⁴/3
Problem 10: (-24x + 42x⁴ + 42x³) ÷ 18x
- -24x⁶ ÷ 18x = -4x⁵/3
- 42x⁴ ÷ 18x = 7x³/3
- 42x³ ÷ 18x = 7x²/3
Answer: -4x⁵/3 + 7x³/3 + 7x²/3
Problem 11: (-40x¹² + 30x¹ - 10x⁸ + 30x⁵ + 80x) 20x²
- -40x¹² ÷ 20x² = -2x¹⁰
- 30x¹ ÷ 20x² = 3x⁸/2 or 1.5x⁸
- -10x ÷ 20x² = -x/2 or -0.5x
- 30x⁵ ÷ 20x² = 3x³/2 or 1.5x³
- 80x ÷ 20x² = 4/x
Answer: -2x¹⁰ + 3x⁸/2 - x/2 + 3x³/2 + 4/x
Problem 12: (77x¹¹ + 33x¹⁰ + 99x⁶ + 88x⁵ + 44x) 11x³
- 77x¹¹ ÷ 11x³ = 7x⁸
- 33x¹⁰ ÷ 11x³ = 3x⁷
- 99x⁶ ÷ 11x³ = 9x³
- 88x⁵ ÷ 11x³ = 8x²
- 44x ÷ 11x³ = 4/x²
Answer: 7x⁸ + 3x⁷ + 9x³ + 8x² + 4/x²
Problem 13: (-40x¹² - 20x¹¹ - 25x⁸ - 30x⁵) ÷ 15x³
- -40x¹² ÷ 15x³ = -8x⁹/3
- -20x¹¹ ÷ 15x³ = -4x⁸/3
- -25x⁸ ÷ 15x³ = -5x⁵/3
- -30x⁵ ÷ 15x³ = -2x²
Answer: -8x⁹/3 - 4x⁸/3 - 5x⁵/3 - 2x²
Final Answer:
1. x³/2 + 5x²/2 + 2x
2. 3x⁵ + 4x² + x
3. 7x³/3 + 5x/3 + 1
4. 8x³ + 9x + 1
5. 2x¹¹ + 2x¹⁰ + x⁷
6. 5x¹⁰ + 2x⁸ + 2x³
7. x⁴ + 3x² + x/3
8. 3x⁸ + 7x⁷/2 + 4x
9. 4x¹⁰/3 - 2x⁵ - 4x⁴/3
10. -4x⁵/3 + 7x³/3 + 7x²/3
11. -2x¹⁰ + 3x⁸/2 - x⁶/2 + 3x³/2 + 4/x
12. 7x + 3x⁷ + 9x³ + 8x² + 4/x²
13. -8x⁹/3 - 4x⁸/3 - 5x⁵/3 - 2x²
Parent Tip: Review the logic above to help your child master the concept of divide a polynomial by a monomial worksheet.
