- Identify the relationship \(W(x) = \frac{A(x)}{L(x)}\) between area, length, and width.
- Use polynomial long division to divide the area \(A(x) = x^4 + 4x^3 + 3x^2 - 4x - 4\) by the length \(L(x) = x^3 + 5x^2 + 8x + 4\).
- Calculate the quotient \(x - 1\), which represents the width of the rectangle.
- Confirm the result against the given options, identifying the width as \(\boxed{x-1}\).
Explanation
- Problem Analysis and Objective
The area of a rectangle is given by the polynomial \(A(x) = x^4 + 4x^3 + 3x^2 - 4x - 4\), and the length is given by \(L(x) = x^3 + 5x^2 + 8x + 4\). Since the relationship between area, length, and width is \(Area = Length \times Width\), the width is the quotient of the division of the area by the length: \(W(x) = \frac{A(x)}{L(x)}\). We need to determine which of the provided options corresponds to this quotient.
- Polynomial Division to Find the Width
To find the width, we perform polynomial long division of \(A(x)\) by \(L(x)\). Dividing the leading term \(x^4\) by \(x^3\) gives the first term of the quotient: \(x\). Multiplying \(x\) by the entire divisor \(L(x)\) yields \(x(x^3 + 5x^2 + 8x + 4) = x^4 + 5x^3 + 8x^2 + 4x\). Subtracting this from \(A(x)\) gives the new dividend:
\((x^4 + 4x^3 + 3x^2 - 4x - 4) - (x^4 + 5x^3 + 8x^2 + 4x) = -x^3 - 5x^2 - 8x - 4\).
Dividing the leading term \(-x^3\) by \(x^3\) gives the second term of the quotient: \(-1\). Multiplying \(-1\) by \(L(x)\) gives \(-(x^3 + 5x^2 + 8x + 4) = -x^3 - 5x^2 - 8x - 4\). Subtracting this from the remaining dividend results in \(0\). Thus, the quotient is \(x - 1\).
- Conclusion and Final Answer
Since the polynomial division of the area \(A(x)\) by the length \(L(x)\) resulted in a quotient of \(x - 1\) with a remainder of \(0\), we have verified that \(A(x) = L(x) \times (x - 1)\). Comparing this result with the provided options, the width of the rectangle is \(x - 1\).
Examples
Polynomial division is a fundamental tool in algebra used for simplifying complex expressions and solving problems involving areas and volumes. In real life, this is often used in architecture and engineering when designing components where the total volume or area of a structure is known and you need to determine the dimensions of its individual parts, such as the width or height of a panel that must fit into a larger design. Mastering this helps in efficiently calculating unknown dimensions in manufacturing or structural planning.