18. The area, in square feet, of a rectangular parking lot is represented by the expression x^3 + 27. Let x + 3 represent the length, in feet, of the parking lot. Which expression represents the width, in feet, of the parking lot?
Answer: A
The width of the parking lot is represented by the expression x² - 3x + 9.
To find the width of the parking lot, we need to factor the area expression x³ + 27 using the given length x + 3. This leads us to the expression for the width being x² - 3x + 9 after applying polynomial division.
A) x² - 3x + 9
This option is correct because when we divide the area expression x³ + 27 by the length expression x + 3, we obtain x² - 3x + 9 as the resulting quotient, which represents the width of the parking lot.
B) x² + 3x + 9
This option is incorrect as it does not match the result from the polynomial division. The presence of the positive 3 in this expression does not align with the necessary calculations derived from factoring x³ + 27.
C) (x + 3)²
This option is also incorrect because (x + 3)² represents the area of a square with side length x + 3, not the width of the rectangular parking lot. The width must be a different expression derived from the area.
D) x² - 9
This option is incorrect since it does not correctly represent the width derived from the area expression. While it can be factored further, it does not correspond to the dimensions required for the parking lot in this context.
E) x + 3
This option is incorrect as it represents the length of the parking lot, not the width. The width must be a distinct expression that results from the area when divided by the length.
Conclusion
The correct answer, x² - 3x + 9, accurately reflects the width of the parking lot derived from the given area expression when divided by the length. All other options fail to provide the correct width, either by misrepresenting the polynomial result or by inaccurately reflecting the dimensions of the parking lot.