Final answer:
The correct quotient when dividing the polynomial (2x³ + 2x² - 6x + 18) by (x + 3) using long division is 2x² - 4x + 6, with a remainder of 0, meaning the division is exact.
Step-by-step explanation:
To divide the polynomial (2x³ + 2x² - 6x + 18) by (x + 3) using long division, begin by setting up the division similar to how you would with numbers.
- Divide the first term of the numerator, 2x³, by the first term of the denominator, x, to get the first term of the quotient, 2x².
- Multiply the entire denominator (x + 3) by the first term of the quotient (2x²) and subtract this from the original polynomial.
- Bring down the next term from the original polynomial and repeat the process until all terms have been accounted for.
- If there is a remainder after all terms have been divided, it can be expressed as a fraction of the original divisor.
Conducting the long division process, we find that the correct quotient is:
2x² - 4x + 6and the remainder is 0, so the division is exact.