Final answer:
To find the height of a hexagonal prism given its volume and the area of its base, divide the volume polynomial by the area polynomial. The result is the height of the prism expressed as a polynomial. In this case, the height is 6x³ - 4x² - 3x - 11.
Step-by-step explanation:
The student has provided the volume of a hexagonal prism as a polynomial (12x⁵ - 8x⁴ - 9x³ - 42x² - 33x - 44) and the area of the base of the prism as another polynomial (2x² + 3x + 4). To find the height of the prism, we need to apply the formula for the volume of a prism, which is V = Ah, where V is the volume, A is the area of the base, and h is the height. By rearranging the formula to solve for the height, we get h = V/A.
By dividing the volume by the area of the base, we obtain an expression for the height:
- The volume given is 12x⁵ - 8x⁴ - 9x³ - 42x² - 33x - 44.
- The area of the base is 2x² + 3x + 4.
- The height h is therefore (12x⁵ - 8x⁴ - 9x³ - 42x² - 33x - 44) / (2x² + 3x + 4), which simplifies to 6x³ - 4x² - 3x - 11.
Therefore, the expression for the height of the prism is 6x³ - 4x² - 3x - 11 (Option a).