Question

The room shown in the figure has a floor space of \(\left(3 x^{2}+8 x+4\right)\) square feet. If the width of the room is \((x+2)\) feet, what is the length?

Short answer

The length of the room is \(3x + 2\) feet.

Step-by-step solution

Analyze the problem

Given the area of the room \(\left(3 x^{2}+8 x+4\right)\) square feet and the width of the room \((x+2)\) feet, we are tasked with finding the length. Since the area of a rectangle is the product of its length and width, the length can be found by dividing the area by the width.

Write down the formula for the area of a rectangle

The formula for the area of a rectangle is Area = Length \times Width. In this case, we can rearrange the formula to solve for length: Length = Area ÷ Width.

Substitute the given values into the formula

Substitute the given values into the formula: Length = \(\left(3 x^{2}+8 x+4\right)\) ÷ \(x + 2\).

Perform the polynomial division

Using polynomial division or synthetic division, we find that \(\left(3 x^{2}+8 x+4\right) \div (x + 2)\) simplifies to \(3x + 2\).

Key concepts

Algebraic Expressions

Algebraic expressions are a fundamental part of algebra, used to represent mathematical phrases that can include numbers, variables, and operations. Key to understanding these expressions are components such as coefficients, variables, and constants. Consider the algebraic expression given in the exercise: \(3x^2 + 8x + 4\). Here, each term consists of:

• Coefficients: These are the numerical parts of the terms, such as 3 and 8.
• Variables: These are letters representing numbers, in this case, \(x\).
• Constant: The number without a variable, here it is 4.

Understanding these elements is crucial in performing operations such as addition, subtraction, and division.
Manipulating algebraic expressions helps solve equations and other algebraic problems. In our problem, dividing \(3x^2 + 8x + 4\) by \(x + 2\) is an example of using algebraic manipulation to find the length of a rectangle.

Rectangular Area

The concept of rectangular area is essential to geometry and algebra. The area of a rectangle is a measure of the space within its sides. It is calculated by multiplying the length by the width. In algebra, when dimensions are given as expressions, understanding area becomes intertwined with algebraic operations.
In our example, the room's rectangular area is given by \(3x^2 + 8x + 4\), while the width is \(x + 2\). To find the unknown dimension, you use the area formula:

• Area = Length \(\times\) Width
• Length = Area \(\div\) Width

Thus, solving for the length involves dividing the total area by the width. This necessitates good knowledge of polynomial division to simplify the area expression to find the corresponding length.

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a linear factor and is particularly useful when dealing with algebraic expressions involving variables. It offers a more efficient alternative to the traditional long division method.
In our exercise, we perform synthetic division to divide the quadratic polynomial \(3x^2 + 8x + 4\) by the linear polynomial \(x + 2\). Here's a general process of synthetic division:

• Choose the divisor: In this case, the divisor is \(x + 2\). Convert it to a form suitable for synthetic division by using \(-2\) (the root of \(x + 2 = 0\)).
• Setup: Write out the coefficients of the dividend polynomial \(3, 8, 4\).
• Perform division: Bring down the first coefficient. Multiply it by the root \(-2\) and write the result underneath the next coefficient. Add together. Repeat this process across all coefficients.

You will end with a new set of coefficients that represents the quotient polynomial, providing the solution to the division problem: \(3x + 2\). This shows us the length of the rectangular room.