Problem 43 - College Algebra

Question

A rectangle with length \(2 x+5\) inches has an area of \(2 x^{4}+15 x^{3}+7 x^{2}-135 x-225\) square inches. Write a polynomial that represents its width.

Step-by-Step Solution

Answer

The polynomial that represents the width of the rectangle is \(x^{3}+5x^{2}-6x-45\)

Step 1: Formulate the Equation for Width

The formula for the area of a rectangle equals length times width (Area = Length*Width). Therefore, the width is equal to the area divided by the length. Write the equation as \(Width = \frac{Area}{Length}\).

Step 2: Substitute the Given Values

Substitute the given values for area and length into the equation obtained in Step 1: \(Width = \frac{2 x^{4}+15 x^{3}+7 x^{2}-135 x-225}{2x+5}\) .

Step 3: Dividing Polynomial

Divide the polynomial in order to reduce it to the simplest form. The result comes as \(x^{3}+5x^{2}-6x-45\).

Key Concepts

Understanding Rectangle Area

Finding the area of a rectangle is a fundamental concept in geometry that helps us determine the size of a space. When you think about the area of a rectangle, it's about figuring out how much flat space it covers. Here, we focus on multiplying the length and width to get the area. The formula to remember is:

• Area = Length × Width

This equation is crucial because it lays the foundation for how we explore other related concepts, like finding unknown dimensions when given certain information. In our example, we started with a known area and a known length, and our task was to solve for an unknown width. By rearranging the formula, we can find the missing value. To find the width when area and length are given, simply divide the area by the length. This shows the vital role of understanding how operations work together in a formula.

Exploring Polynomial Expressions

A polynomial is an expression that involves variables raised to various powers, combined using addition, subtraction, and sometimes multiplication. In the given exercise, each component of the rectangle's area is part of a polynomial expression. This might look complex initially, but it’s a structured and logical way to represent numbers and operations.
Consider the polynomial in our exercise:

• The area expression is: \[2x^{4} + 15x^{3} + 7x^{2} - 135x - 225\]
• The length expression is: \[2x + 5\]

Polynomials like these are often used in algebra due to their versatility in expressing relationships in terms of variables. Understanding how to manipulate these expressions is essential for solving the problem and involves techniques like polynomial division, which simplifies the process and makes solving for unknowns feasible.

Width Calculation from Area and Length Using Polynomial Division

Calculating the width of a rectangle when given its area and length requires a logical process, specifically when dealing with polynomials. We performed polynomial division to find the width.
Here's how it works:

• Start with the equation for width: \[\text{Width} = \frac{\text{Area}}{\text{Length}}\]
• Substitute the given polynomial expressions for the area and length into this equation.
• This means you will divide: \[\frac{2x^{4} + 15x^{3} + 7x^{2} - 135x - 225}{2x + 5}\]

Polynomial division simplifies this expression by breaking it down, resulting in:

• The calculated width becomes: \[x^{3} + 5x^{2} - 6x - 45\]

Understanding polynomial division is fundamental since it helps find simpler expressions or solve equations, representing real-world dimensions or measurements like the width in our context.