Problem 35 - College Algebra with Corequisite Support

Question

For the following exercises, multiply the binomials. \((4+4 m)(4-4 m)\)

Step-by-Step Solution

Answer

The product is \(16 - 16m^2\).

Step 1: Recognize the Pattern

Notice the expression is of the form \((a+b)(a-b)\), where \(a = 4\) and \(b = 4m\). This pattern is known as the difference of squares.

Step 2: Apply the Difference of Squares Identity

The difference of squares identity states that \((a+b)(a-b) = a^2 - b^2\). Use this to rewrite the expression: \((4)^2 - (4m)^2\).

Step 3: Calculate the Squares

Calculate \((4)^2 = 16\) and \((4m)^2 = 16m^2\). Substitute these values back into the expression.

Step 4: Final Result

Substitute the values into the equation from Step 2: \[16 - 16m^2\]. Thus, the product of the binomials is \(16 - 16m^2\).

Key Concepts

Binomials

A binomial is a simple algebraic expression that consists of two terms. These terms are usually separated by a plus or a minus sign. In algebra, binomials are fundamental building blocks because they frequently appear in equations and algebraic identities. For example, in the expression

• (4 + 4m)
• (4 - 4m)

both are binomials as they have exactly two terms. Working with binomials often involves operations like addition, subtraction, multiplication, and division, and understanding these operations is essential for solving complex algebraic problems effectively.
Understanding how to manipulate binomials, like rewriting or expanding them, allows you to simplify expressions and solve equations. In this exercise, multiplying the binomials involves recognizing specific algebraic patterns, such as the difference of squares, to solve it efficiently. Recognizing these patterns makes working with binomials quicker and easier.

Difference of Squares

The difference of squares is a handy algebraic pattern. It states that multiplying two binomials of the form

• (a + b)
• (a - b)

results in the expression

• a² - b²

This identity highlights the elimination of the middle terms when these specific kinds of binomials are multiplied, greatly simplifying the process. In our exercise, given the pair

• (4 + 4m)
• (4 - 4m)

we can apply the difference of squares identity directly. The values of "a" and "b" in our context are 4 and 4m, respectively. Therefore, you compute

• a² = 4² = 16
• b² = (4m)² = 16m²

This leads to the final simplified result:

• 16 - 16m²

. Recognizing and applying the difference of squares saves time and effort because it provides a straightforward path to the solution.

Algebraic Identities

Algebraic identities are equations that are universally true and hold for any values of the involved variables. They serve as shortcuts in algebra, allowing us to simplify complex expressions without needing to perform detailed calculations every time. The difference of squares

• (a + b)(a - b) = a² - b²

is one such well-known identity. Identifying that a problem fits into an algebraic identity can drastically shorten the time required to find a solution, offering a clear method to follow.
In our specific problem, realizing that it is a difference of squares made multiplying the binomials

• (4 + 4m)
• (4 - 4m)

easier by allowing us to directly apply the identity, skipping over additional steps of traditional multiplication. Knowing these identities enhances your mathematical toolkit, making it easier to tackle a variety of algebraic challenges with confidence. Mastery of these identities enables efficient and effective problem-solving in algebra.