The area of a rectangle is $(x^4+4 x^3+3 x^2-4 x-4)$, and the length of the rectangle is $(x^3+5 x^2+8 x+4)$. If area = length $\times$ width, what is the width of the rectangle?
A. $x+1$
B. $x-9$
C. $x+4$
D. $x-1$
Answer (1)
Divide the area of the rectangle by its length to find the width.
Perform polynomial division: x 3 + 5 x 2 + 8 x + 4 x 4 + 4 x 3 + 3 x 2 − 4 x − 4 .
Simplify the expression to obtain the width.
The width of the rectangle is x − 1 .
Explanation
Problem Analysis We are given the area and the length of a rectangle and asked to find the width. We know that Area = Length × Width, so Width = Area / Length. In this case, the area A = x 4 + 4 x 3 + 3 x 2 − 4 x − 4 and the length L = x 3 + 5 x 2 + 8 x + 4 . Therefore, the width W = L A = x 3 + 5 x 2 + 8 x + 4 x 4 + 4 x 3 + 3 x 2 − 4 x − 4 .
Solution Strategy To find the width, we need to divide the polynomial representing the area by the polynomial representing the length. This can be done using polynomial long division or by factoring both polynomials and canceling common factors.
Polynomial Division Let's perform the polynomial division: x 3 + 5 x 2 + 8 x + 4 x 4 + 4 x 3 + 3 x 2 − 4 x − 4 After performing the polynomial division (or using sympy to simplify the expression), we find that the result is x − 1 .
Final Answer Therefore, the width of the rectangle is x − 1 .
Examples
Understanding polynomial division is crucial in various fields, such as engineering and computer science. For instance, when designing a communication system, engineers use polynomials to model signals. Dividing one polynomial by another can help in decoding or filtering signals, ensuring clear communication. Similarly, in computer graphics, polynomial division can be used to manipulate curves and surfaces, allowing for realistic rendering of 3D objects. This problem demonstrates a fundamental algebraic technique with broad applications.
