Final answer:
The result of dividing 9x³ + 12x² + 27x + 8 by 3x + 1 is 3x² + 3x + 8, found using polynomial long division. The steps include dividing term by term and subtracting the product from the dividend, yielding the quotient without any remainder.
Step-by-step explanation:
The result when 9x³ + 12x² + 27x + 8 is divided by 3x + 1 can be found using either polynomial long division or synthetic division. For polynomial long division, we divide the first term of the dividend (9x³) by the first term of the divisor (3x), to get 3x². Multiplying the divisor by 3x² and subtracting from the dividend gives us a new polynomial. We then repeat the process with the new polynomial until the terms cannot be divided by 3x anymore. The remainder will be a constant since it is lower in degree than the divisor.
Following this process for our given polynomial gets us:
- Step 1: 9x³ divided by 3x is 3x².
- Step 2: 3x × 3x² is 9x³ and 1 × 3x² is 3x². Subtracting these from the dividend gives 12x² - 3x² which is 9x².
- Step 3: 9x² divided by 3x is 3x.
- Step 4: 3x × 3x is 9x² and 1 × 3x is 3x. Subtracting these from the new polynomial part 9x² + 27x gives 24x.
- Step 5: 24x divided by 3x is 8.
- Step 6: 3x × 8 is 24x and 1 × 8 is 8. Subtracting these from the remaining polynomial part 24x + 8 gives 0, indicating no remainder.
Therefore, the quotient is 3x² + 3x + 8, which corresponds to option (a).