- Identify the dividend coefficients as \(1, 2, -17, -36\) and the root as \(-4\).
- Perform the synthetic division by carrying down the leading coefficient and iteratively multiplying by the root and adding to the next coefficient.
- Obtain the quotient coefficients \(1, -2, -9\), which form the quadratic expression \(x^2 - 2x - 9\).
- The final expression for the area of the base is \(\boxed{x^2 - 2x - 9}\).
Explanation
- Problem Analysis and Setup
The volume \(V\) of a rectangular prism is given by the product of its base area \(A_b\) and its height \(h\). We are given the volume \(V(x) = x^3 + 2x^2 - 17x - 36\) and the height \(h(x) = x + 4\). The relationship is \(V = A_b \times h\), so to find the area of the base, we must perform the polynomial division:
\[A_b(x) = \frac{x^3 + 2x^2 - 17x - 36}{x + 4}\]
We will use synthetic division to compute this quotient.
- Performing Synthetic Division
To perform the synthetic division for the expression \(\frac{x^3 + 2x^2 - 17x - 36}{x + 4}\), we first identify the coefficients of the dividend polynomial \(x^3 + 2x^2 - 17x - 36\), which are \(1, 2, -17,\) and \(-36\). The divisor is in the form \(x - c\), where \(x - (-4) = x + 4\), so the root \(c = -4\). We set up the synthetic division tableau as follows:
\(-4 \ \vline \ 1 \quad \ 2 \quad -17 \quad -36\)
\(\quad \vline \quad \quad -4 \quad \ \ 8 \quad \ \ \ 36\)
\(\quad \vline \overline{ \quad 1 \quad -2 \quad \ \ -9 \quad \ \ \ \ \ 0 }\)
- First, bring down the leading coefficient \(1\).
- Multiply by the root \(-4\): \(1 \times (-4) = -4\). Add this to the next coefficient \(2\): \(2 + (-4) = -2\).
- Multiply by the root \(-4\): \(-2 \times (-4) = 8\). Add this to the next coefficient \(-17\): \(-17 + 8 = -9\).
- Multiply by the root \(-4\): \(-9 \times (-4) = 36\). Add this to the last coefficient \(-36\): \(-36 + 36 = 0\).
The quotient coefficients are \(1, -2,\) and \(-9\), which represent the polynomial \(x^2 - 2x - 9\).
- Conclusion and Result
The result of dividing \(x^3 + 2x^2 - 17x - 36\) by \(x + 4\) is the quadratic polynomial \(x^2 - 2x - 9\). This represents the area of the base \(A_b(x)\). Comparing this to the given options, we see that it matches the third option provided. Therefore, the area of the base is \(x^2 - 2x - 9\).
Examples
In engineering and manufacturing, determining dimensions from volume and height is a common task. For instance, if you are designing a storage box and you know the total volume required for the products inside and the height of the shelves in your warehouse, you can use polynomial division to determine the necessary surface area of the base of your boxes to ensure they fit correctly on the shelves. This mathematical approach helps in optimizing storage space and ensuring efficient logistics and product packaging.