Problem 82 - Algebra A Combined Function

Question

Express each as a product of polynomials in \(x .\) Then multiply and simplify. Find the area of the rectangular canvas if its length is \((3 x-2)\) inches and its width is \((x-4)\) inches.

Step-by-Step Solution

Answer

The area of the rectangular canvas is \(3x^2 - 14x + 8\) square inches.

Step 1: Express the Area as a Polynomial Expression

The area of a rectangle is calculated as the product of its length and width. In this case, the length is \((3x - 2)\) inches and the width is \((x - 4)\) inches. Therefore, the area \(A\) can be expressed as a polynomial: \(A = (3x - 2)(x - 4)\).

Step 2: Expand the Polynomial Expression

To find the product, use the distributive property (also known as the FOIL method for binomials). Multiply each term in the first binomial by each term in the second binomial:- Multiply \(3x\) by \(x\) to get \(3x^2\).- Multiply \(3x\) by \(-4\) to get \(-12x\).- Multiply \(-2\) by \(x\) to get \(-2x\).- Multiply \(-2\) by \(-4\) to get \(+8\).So, the expanded expression is \(3x^2 - 12x - 2x + 8\).

Step 3: Simplify the Polynomial Expression

Combine like terms in the expanded expression:- The terms \(-12x\) and \(-2x\) combine to form \(-14x\).Thus, the polynomial simplifies to \(3x^2 - 14x + 8\).

Key Concepts

Area of a Rectangle

To calculate the area of a rectangle, we multiply its length by its width. This formula:

• is a fundamental concept in geometry
• helps us determine how much space the rectangle covers

In the given exercise, the length is \((3x - 2)\) inches and the width is \((x - 4)\) inches. Thus, the area of the rectangle becomes \((3x - 2)(x - 4)\). By understanding this concept, you can solve many real-world problems involving rectangular areas. Learning to express the area in terms of polynomials is an essential skill, especially when dealing with variable dimensions.

Distributive Property

The distributive property is a key principle in algebra that simplifies expressions where terms are distributed over addition or subtraction within parentheses. It's fundamental for multiplying polynomials and variables. The formula for the distributive property is:

• \(a(b + c) = ab + ac\)

In the context of our rectangle area, we apply this property. Each term of the first binomial \((3x - 2)\) is multiplied with each term of the second binomial \((x - 4)\):

• First, multiply \(3x\) by \(x\) resulting in \(3x^2\)
• Then, multiply \(3x\) by \(-4\) resulting in \(-12x\)
• Next, multiply \(-2\) by \(x\) resulting in \(-2x\)
• Finally, multiply \(-2\) by \(-4\) resulting in \(+8\)

This method ensures that all terms are accounted for and simplified accordingly.

Binomial Expansion

Binomial expansion refers to multiplying two binomials, which means expanding an expression that involves binomials to its full polynomial form. This concept often utilizes the distributive property. For example, the expression \((3x - 2)(x - 4)\) can be expanded:

• First, we apply the distributive property for each term, as outlined previously.
• This results in: \(3x^2 - 12x - 2x + 8\)

Once expanded, the next task is to simplify by combining like terms. Here, \(-12x\) and \(-2x\) are combined to form \(-14x\). Thus, the complete expanded and simplified form is \(3x^2 - 14x + 8\). Binomial expansion helps break down complex expressions and is used extensively in algebra to simplify and solve polynomial equations.