Answer: To divide 9x³ - 18x² - x + 2 by 3x + 1 using long division, we can follow the following steps:
Step 1: Divide the first term of the dividend 9x³ by the first term of the divisor 3x. The result is 3x².
Step 2: Multiply the divisor 3x + 1 by 3x². The result is 9x³ + 3x².
Step 3: Subtract 9x³ + 3x² from 9x³ - 18x² - x + 2. The result is -18x² - 4x + 2.
Step 4: Repeat the division and multiplication process by dividing the first term of the new dividend, -18x², by the first term of the divisor, 3x. The result is -6x.
Step 5: Multiply the divisor 3x + 1 by -6x. The result is -18x + 6.
Step 6: Subtract -18x + 6 from -18x² - 4x + 2. The result is -4x² + 10.
Step 7: Repeat the division and multiplication process by dividing the first term of the new dividend, -4x², by the first term of the divisor, 3x. The result is -4/3 x.
Step 8: Multiply the divisor 3x + 1 by -4/3 x. The result is -4/3 x + 4/3.
Step 9: Subtract -4/3 x + 4/3 from -4x² + 10. The result is 10 - 4/3.
Thus, the final answer is:
(9x³ - 18x² - x + 2) / (3x + 1) = 3x² - 6x - 4/3 x + 4/3 + 10
Explanation: