Problem 81 - Algebra A Combined Function

Question

Express each as a product of polynomials in \(x .\) Then multiply and simplify. Find the area of the square rug if its side is \((2 x+1)\) feet.

Step-by-Step Solution

Answer

The area of the square rug is \(4x^2 + 4x + 1\) square feet.

Step 1: Recall the Formula for Area Calculation

The area of a square is given by the formula \( ext{Area} = ext{side} imes ext{side} \). In this case, the side of the square is \((2x + 1)\). Therefore, the area can be expressed as \((2x + 1) \times (2x + 1)\).

Step 2: Expand the Expression

To find the area, we need to expand the expression \((2x + 1) \times (2x + 1)\) which is essentially \((2x + 1)^2\). We can expand this using the distributive property (or FOIL method for binomials):\[(2x + 1) \cdot (2x + 1) = 2x(2x) + 2x(1) + 1(2x) + 1(1)\]

Step 3: Calculate Each Term

Let's compute each term from the expansion:1. \(2x \cdot 2x = 4x^2\)2. \(2x \cdot 1 = 2x\)3. \(1 \cdot 2x = 2x\)4. \(1 \cdot 1 = 1\)

Step 4: Combine Like Terms

Add the calculated terms together:\[4x^2 + 2x + 2x + 1 = 4x^2 + 4x + 1\].

Step 5: Simplify the Expression

Combine any like terms if needed (in this case, \(2x\) and \(2x\) are combined in the previous step) to obtain the simplified expression for the area of the square:\[4x^2 + 4x + 1\].

Key Concepts

Area of a Square

The area of a square is an important concept in geometry. It's the amount of space contained within its boundaries. Calculating the area helps us understand how much surface area the square will cover. The formula for computing the area of a square is fairly straightforward: it is the square of its side length.
So, if you know the length of a square's side, calculating the area is just a matter of squaring that number.
Here's the formula:

• Area of a square = side × side.

In this example, if the side of the square is given as \((2x + 1)\), the area will be calculated as \((2x + 1)\) times \((2x + 1)\). This expression introduces us to the concept of expanding binomials to find the area.

Distributive Property

The distributive property is a fundamental concept in algebra that simplifies multiplication involving addition. It states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately.
The formula for the distributive property is:

• \(a(b + c) = ab + ac\)

In the context of multiplying binomials like \((2x + 1)(2x + 1)\), we use the distributive property to break down the multiplication. Each term in the first binomial is distributed across each term in the second binomial. This distribution helps in isolating each product term, making it easier to simplify later on. Understanding the distributive property is essential for handling polynomial operations.

FOIL Method

The FOIL method is a handy mnemonic for multiplying two binomials. FOIL stands for First, Outer, Inner, and Last, representing the pairs of terms you multiply together when expanding the product of two binomials.
Let's see how it works using our example, \((2x + 1)(2x + 1)\):

• **First:** Multiply the first terms: \(2x imes 2x = 4x^2\)
• **Outer:** Multiply the outer terms: \(2x imes 1 = 2x\)
• **Inner:** Multiply the inner terms: \(1 imes 2x = 2x\)
• **Last:** Multiply the last terms: \(1 imes 1 = 1\)

After finding each product, you combine them to get the expanded expression. This simplicity and clarity make the FOIL method a popular choice for expanding binomials.

Expanding Binomials

Expanding binomials is a crucial skill in algebra that involves multiplying and simplifying expressions. When you expand a binomial, you use methods like distributive property or FOIL to remove the parentheses and combine like terms.
The expanded form provides a clearer representation of the terms without parentheses, making it easier to perform further algebraic operations.
Using the previously mentioned example, \((2x + 1)(2x + 1)\), the expansion process yields: \[4x^2 + 2x + 2x + 1\].
Once you've calculated each term, you combine like terms to simplify. This simplification gives you the tidy form: \[4x^2 + 4x + 1\].
Expanding a binomial expression this way makes it easier for further analysis and operations such as addition or subtraction with other polynomials.