The area of a rectangle is [tex]$(x^3-5 x^2+3 x-15)$[/tex], and the width of the rectangle is [tex]$(x^2+3)$[/tex]. If area = length [tex]$\times$[/tex] width, what is the length of the rectangle?
A. [tex]$x+5$[/tex]
B. [tex]$x-15$[/tex]
C. [tex]$x+15$[/tex]
D. [tex]$x-5$[/tex]

The length of the rectangle can be calculated using the formula Length = Area ÷ Width. After factoring the area and simplifying the expression, we find the length to be x − 5 . Therefore, the correct answer is D. x − 5 .
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Answered by Anonymous | 2025-07-04

Express the length as a fraction: L = 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 ) .
Substitute the factored form: L = x 2 + 3 ( x 2 + 3 ) ( x − 5 ) ​ .
Simplify by canceling the common factor: L = x − 5 . The length of the rectangle is x − 5 ​ .

Explanation

Understanding the Problem We are given the area of a rectangle as x 3 − 5 x 2 + 3 x − 15 and the width as x 2 + 3 . We need to find the length of the rectangle, knowing that Area = Length × Width.

Setting up the Equation Let A be the area, L be the length, and W be the width. We have:


A = x 3 − 5 x 2 + 3 x − 15
W = x 2 + 3
Since A = L × W , we can find the length by dividing the area by the width: L = W A ​

Expressing Length as a Fraction So, we have:

L = x 2 + 3 x 3 − 5 x 2 + 3 x − 15 ​

Factoring the Numerator To simplify this expression, we can factor the numerator (the area expression) by grouping:

x 3 − 5 x 2 + 3 x − 15 = x 2 ( x − 5 ) + 3 ( x − 5 ) = ( x 2 + 3 ) ( x − 5 )

Substituting the Factored Form Now we substitute the factored form back into the expression for the length:

L = x 2 + 3 ( x 2 + 3 ) ( x − 5 ) ​

Simplifying the Expression We can now cancel the common factor of ( 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 polynomial factorization and division is crucial in various fields, such as engineering and computer graphics. For instance, when designing a rectangular garden with a specific area and width defined by polynomial expressions, determining the length involves polynomial division. If the area of the garden is given by x 3 − 5 x 2 + 3 x − 15 square feet and the width is x 2 + 3 feet, finding the length ( x − 5 ) ensures accurate planning and resource allocation for the garden's layout.

Answered by GinnyAnswer | 2025-07-04