divide the polynomials using long division method

1.) 2x³-9x²+15 ÷ 2x-5

2.) x⁴+5x³+10x²-9x-34÷x+2

Answer:sanaStep-by-step explanation:1.) 2x³ - 9x² + 15 ÷ 2x - 5 1. Set up the long division: ____________2x - 5 | 2x³ - 9x² + 0x + 15  2. Divide the first term of the dividend (2x³) by the first term of the divisor (2x): x²____________2x - 5 | 2x³ - 9x² + 0x + 15 3. Multiply the quotient (x²) by the divisor (2x - 5): x²____________2x - 5 | 2x³ - 9x² + 0x + 15 2x³ - 5x² 4. Subtract the result from the dividend: x²____________2x - 5 | 2x³ - 9x² + 0x + 15 2x³ - 5x² --------- -4x² + 0x 5. Bring down the next term (0x): x²____________2x - 5 | 2x³ - 9x² + 0x + 15 2x³ - 5x² --------- -4x² + 0x 6. Divide the first term of the new dividend (-4x²) by the first term of the divisor (2x): x² - 2x________2x - 5 | 2x³ - 9x² + 0x + 15 2x³ - 5x² --------- -4x² + 0x 7. Multiply the new quotient (-2x) by the divisor (2x - 5): x² - 2x________2x - 5 | 2x³ - 9x² + 0x + 15 2x³ - 5x² --------- -4x² + 0x -4x² + 10x 8. Subtract the result from the new dividend: x² - 2x________2x - 5 | 2x³ - 9x² + 0x + 15 2x³ - 5x² --------- -4x² + 0x -4x² + 10x --------- -10x + 15 9. Bring down the next term (15): x² - 2x________2x - 5 | 2x³ - 9x² + 0x + 15 2x³ - 5x² --------- -4x² + 0x -4x² + 10x --------- -10x + 15 10. Divide the first term of the new dividend (-10x) by the first term of the divisor (2x): x² - 2x - 5_____2x - 5 | 2x³ - 9x² + 0x + 15 2x³ - 5x² --------- -4x² + 0x -4x² + 10x --------- -10x + 15 11. Multiply the new quotient (-5) by the divisor (2x - 5): x² - 2x - 5_____2x - 5 | 2x³ - 9x² + 0x + 15 2x³ - 5x² --------- -4x² + 0x -4x² + 10x --------- -10x + 15 -10x + 25 12. Subtract the result from the new dividend: x² - 2x - 5_____2x - 5 | 2x³ - 9x² + 0x + 15 2x³ - 5x² --------- -4x² + 0x -4x² + 10x --------- -10x + 15 -10x + 25 --------- -10  The quotient is x^2 - 2x - 5 and the remainder is -10. Therefore, \frac{2x^3 - 9x^2 + 15}{2x - 5} = x^2 - 2x - 5 - \frac{10}{2x - 5}.   2.) x⁴ + 5x³ + 10x² - 9x - 34 ÷ x + 2 1. Set up the long division: ________________x + 2 | x⁴ + 5x³ + 10x² - 9x - 34 2. Divide the first term of the dividend (x⁴) by the first term of the divisor (x): x³_______________x + 2 | x⁴ + 5x³ + 10x² - 9x - 34 3. Multiply the quotient (x³) by the divisor (x + 2): x³_______________x + 2 | x⁴ + 5x³ + 10x² - 9x - 34 x⁴ + 2x³ 4. Subtract the result from the dividend: x³_______________x + 2 | x⁴ + 5x³ + 10x² - 9x - 34 x⁴ + 2x³ ------- 3x³ + 10x² 5. Bring down the next term (-9x): x³_______________x + 2 | x⁴ + 5x³ + 10x² - 9x - 34 x⁴ + 2x³ ------- 3x³ + 10x² - 9x 6. Divide the first term of the new dividend (3x³) by the first term of the divisor (x): x³ + 3x²__________x + 2 | x⁴ + 5x³ + 10x² - 9x - 34 x⁴ + 2x³ ------- 3x³ + 10x² - 9x 7. Multiply the new quotient (3x²) by the divisor (x + 2): x³ + 3x²__________x + 2 | x⁴ + 5x³ + 10x² - 9x - 34 x⁴ + 2x³ ------- 3x³ + 10x² - 9x 3x³ + 6x² 8. Subtract the result from the new dividend: x³ + 3x²__________x + 2 | x⁴ + 5x³ + 10x² - 9x - 34 x⁴ + 2x³ ------- 3x³ + 10x² - 9x 3x³ + 6x² ------- 4x² - 9x 9. Continue this process until you reach the remainder. The final result will be:x^3 + 3x^2 + 4x - 17 The quotient is x³ + 3x² + 4x - 17 and the remainder is 0. Therefore, \frac{x^4 + 5x^3 + 10x^2 - 9x - 34}{x + 2} = x^3 + 3x^2 + 4x - 17.

Answered by maryrosejoydiasana | 2025-07-28