I'll solve each polynomial division problem step by step.
Problem 1: (x² - 2x - 11) ÷ (x - 5)
This is polynomial long division. Let me work through it:
- x² x = x
- Multiply: x(x - 5) = x² - 5x
- Subtract: (x² - 2x - 11) - (x² - 5x) = 3x - 11
- 3x ÷ x = 3
- Multiply: 3(x - 5) = 3x - 15
- Subtract: (3x - 11) - (3x - 15) = 4
Answer: x + 3 with remainder 4, or x + 3 + 4/(x-5)
Problem 2: (16x⁴ + 12x³ + 2x²) ÷ 4x²
Divide each term by 4x²:
- 16x⁴ ÷ 4x² = 4x²
- 12x³ 4x² = 3x
- 2x² ÷ 4x² = 1/2
Answer: 4x² + 3x + 1/2
Problem 3: (x³ + 5x² + 4x) ÷ 10x²
Divide each term by 10x²:
- x³ ÷ 10x² = x/10
- 5x² ÷ 10x² = 5/10 = 1/2
- 4x ÷ 10x² = 4/(10x) = 2/(5x)
Answer: x/10 + 1/2 + 2/(5x)
Problem 4: (10x³ + 2x² + 5x) ÷ 10x²
Divide each term by 10x²:
- 10x³ ÷ 10x² = x
- 2x² ÷ 10x² = 2/10 = 1/5
- 5x ÷ 10x² = 5/(10x) = 1/(2x)
Answer: x + 1/5 + 1/(2x)
Problem 5: (4x⁴ + 5x³ + 20x²) ÷ 10x²
Divide each term by 10x²:
- 4x⁴ ÷ 10x² = 4x²/10 = 2x²/5
- 5x³ ÷ 10x² = 5x/10 = x/2
- 20x² ÷ 10x² = 2
Answer: 2x²/5 + x/2 + 2
Problem 6: (24x⁴ + 16x³ + 2x²) 8x³
Divide each term by 8x³:
- 24x⁴ ÷ 8x³ = 3x
- 16x³ ÷ 8x³ = 2
- 2x² ÷ 8x³ = 2/(8x) = 1/(4x)
Answer: 3x + 2 + 1/(4x)
Final Answer:
1. x + 3 + 4/(x-5)
2. 4x² + 3x + 1/2
3. x/10 + 1/2 + 2/(5x)
4. x + 1/5 + 1/(2x)
5. 2x²/5 + x/2 + 2
6. 3x + 2 + 1/(4x)
Parent Tip: Review the logic above to help your child master the concept of divide a polynomial by a monomial worksheet.
