The area of a rectangle is $(x^3-5x^2+3x-15)$, and the width of the rectangle is $(x^2+3)$. If area = length $\times$ width, what is the length of the rectangle?
A. $x+5$
B. $x-15$
C. $x+15$
D. $x-5$
Answer (2)
Express the length as L = W A = x 2 + 3 x 3 − 5 x 2 + 3 x − 15 .
Factor the numerator by grouping: x 3 − 5 x 2 + 3 x − 15 = ( x 2 + 3 ) ( x − 5 ) .
Cancel the common factor ( x 2 + 3 ) from the numerator and denominator.
The length of the rectangle is L = x − 5 , so the answer is x − 5 .
Explanation
Problem Analysis We are given the area of a rectangle as A = x 3 − 5 x 2 + 3 x − 15 and the width as W = x 2 + 3 . We need to find the length L of the rectangle, knowing that A = L × W .
Finding the Length To find the length, we can rearrange the formula to L = W A . So, we have L = x 2 + 3 x 3 − 5 x 2 + 3 x − 15 .
Factoring the Numerator Now, we need to simplify the expression. Let's factor the numerator x 3 − 5 x 2 + 3 x − 15 . We can use factoring by grouping: x 3 − 5 x 2 + 3 x − 15 = x 2 ( x − 5 ) + 3 ( x − 5 ) = ( x 2 + 3 ) ( x − 5 ) .
Substituting Back Substitute the factored form back into the expression for L : L = x 2 + 3 ( x 2 + 3 ) ( x − 5 ) .
Simplifying the Expression Now, we can cancel the common factor ( x 2 + 3 ) from the numerator and the denominator: L = x − 5 .
Final Answer Therefore, the length of the rectangle is x − 5 .
Examples
Understanding the area and dimensions of rectangles is crucial in various real-world applications. For instance, when designing a room layout, you need to determine the length of a room given its area and width. If the area of a rectangular room is represented by the expression x 3 − 5 x 2 + 3 x − 15 square feet and the width is x 2 + 3 feet, then finding the length using polynomial division or factoring, as we did, helps determine if furniture or other items will fit properly. This ensures efficient use of space and accurate planning in interior design or architecture.
The length of the rectangle, given the area and width, is found by dividing the area A by the width W . By factoring the area, we simplify to find that the length is x − 5 . Thus, the correct answer is option D: x − 5 .
;
