Problem 25 - Algebra 1: Concepts and Skills

Question

Write the product of the sum and difference. $$ (2 m+2)(2 m-2) $$

Step-by-Step Solution

Answer

The product of the sum and difference of the expression \((2m+2)(2m-2)\) is \(4m^2 - 4\).

Step 1: Identify the Terms

Identify the terms in the provided expression. Here, \(a = 2m\) and \(b = 2\). The exercise has the form \((a+b)(a-b)\), which follows the pattern of difference of squares.

Step 2: Applying the Difference of Squares

Apply the differences of squares formula \((a+b)(a-b) = a^2 - b^2\). Substitute \(a = 2m\) and \(b = 2\), yielding \((2m)^2 - 2^2\).

Step 3: Simplification

Simplify the expression. \(4m^2 - 4\) is the simplified expression resulting from subtracting squares.

Key Concepts

Difference of Squares

The difference of squares is a valuable algebraic pattern that simplifies certain expressions. It comes in the form

• \((a + b)(a - b) = a^2 - b^2\)

This pattern identifies that when two binomials are multiplied, one being the sum and the other the difference of the same terms, it results in the difference between the square of these terms.
In the given exercise, the expressions

• \((2m + 2)\)
• \((2m - 2)\)

represent this pattern. Recognizing that the terms are structured this way allows us to quickly apply the difference of squares formula to achieve a simplified result.

Simplification in Algebra

Simplification in algebra is the process of reducing an expression to its most concise form. This process often involves combining like terms and applying algebraic identities.
When dealing with a difference of squares, like in the original exercise, you utilize the identity to condense the product of binomials. After applying the difference of squares formula,

• \(a^2 - b^2 = (2m)^2 - 2^2\)
• \(4m^2 - 4\)

This demonstrates the simplification by computing the squares of each term inside the parentheses.
The expression reduces neatly, highlighting the power of recognizing specific patterns and identities in algebra to streamline calculations.

Binomial Product

The binomial product involves multiplying two terms, forming an expression that can often follow recognizable patterns like the difference of squares.
Here, the binomial product

• \((2m + 2)\)
• \((2m - 2)\)

translates through multiplication into a wider algebraic scenario via

• \((a + b)(a - b)\)

Expressing this product in terms of known algebraic identities saves time and reduces complexity. It's essential to master these repetitive patterns because they frequently occur in algebra, making problem-solving more efficient and intuitive.