- Identify the volume \(V(x) = 2x^3 + 9x^2 - 8x - 36\) and height \(h(x) = x + 2\).
- Recognize that the base area \(A(x) = \frac{V(x)}{h(x)}\).
- Apply synthetic division with root \(k = -2\) to coefficients \([2, 9, -8, -36]\).
- Obtain the resulting quotient \(A(x) = 2x^2 + 5x - 18\), corresponding to option c.
- The base area is \(\boxed{2x^2 + 5x - 18}\).
Explanation
- Analysis and Problem Setup
The volume of a rectangular prism is given by the product of its base area \(A(x)\) and its height \(h(x)\). We are given the volume \(V(x) = 2x^3 + 9x^2 - 8x - 36\) and the height \(h(x) = x + 2\). To find the base area, we perform the polynomial division \(\frac{V(x)}{h(x)} = \frac{2x^3 + 9x^2 - 8x - 36}{x + 2}\). Since the divisor is in the form \(x - k\) with \(k = -2\), we can use synthetic division.
- Performing Synthetic Division
We set up the synthetic division using the coefficients of \(V(x)\) (2, 9, -8, -36) and the root \(k = -2\).
- Bring down the first coefficient: 2.
- Multiply 2 by -2 to get -4. Add this to 9 to get 5.
- Multiply 5 by -2 to get -10. Add this to -8 to get -18.
- Multiply -18 by -2 to get 36. Add this to -36 to get 0.
The result row is 2, 5, -18, representing the coefficients of the quadratic polynomial \(A(x) = 2x^2 + 5x - 18\).
- Conclusion and Final Result
By performing the synthetic division, we have successfully divided the polynomial \(V(x) = 2x^3 + 9x^2 - 8x - 36\) by \(x + 2\). The resulting quotient is the area of the base \(A(x) = 2x^2 + 5x - 18\). Comparing this result with the provided options, it matches option c. Therefore, the area of the base is \(\boxed{2x^2 + 5x - 18}\).
Examples
In engineering and architecture, calculating the dimensions of objects is essential. If you know the total volume of a structure, such as a concrete foundation or a storage bin, and you know one of its dimensions (like height or width), you can determine the other unknown area or length using polynomial division. This application helps professionals ensure structural efficiency and accurate material estimation for construction projects.