Problem 55 - Algebra A Combined Function

Question

Multiply. $$ (2 m+5 n)(2 m-5 n) $$

Step-by-Step Solution

Answer

\(4m^2 - 25n^2\)

Step 1: Recognize the Pattern

Notice that the expression \((2m + 5n)(2m - 5n)\) follows the difference of squares pattern \((a + b)(a - b) = a^2 - b^2\). Here, \(a = 2m\) and \(b = 5n\).

Step 2: Apply the Difference of Squares Formula

Using the difference of squares formula, we substitute \(a = 2m\) and \(b = 5n\) into \(a^2 - b^2\).

Step 3: Square the First Term

Calculate \((2m)^2\). This gives: \[(2m)^2 = 4m^2.\]

Step 4: Square the Second Term

Calculate \((5n)^2\). This gives: \[(5n)^2 = 25n^2.\]

Step 5: Subtract the Squares

Now subtract the second squared term from the first squared term to get the final answer: \[4m^2 - 25n^2.\]

Key Concepts

Algebraic Expressions

Algebraic expressions are a foundational concept in mathematics. They consist of variables, constants, and operations that together create phrases that express algebraic relationships. In the exercise, we have an expression involving the terms \(2m\) and \(5n\), which are both algebraic expressions on their own. Here:

• \(2m\) is the product of 2 and the variable \(m\).
• \(5n\) is the product of 5 and the variable \(n\).

Both terms are combined using the operations of addition and subtraction.
Expressions such as (2m + 5n) and (2m - 5n) are seen in various algebraic operations including polynomial multiplication.Understanding these expressions is key to performing operations like the difference of squares. By recognizing the components, we effectively manipulate and simplify expressions in equations.

Polynomial Multiplication

Polynomial multiplication involves distributing each term in one polynomial by every term in another polynomial. Here, we use a specific case, known as the difference of squares. The expression offered in the exercise (2m + 5n)(2m - 5n)can be multiplied using this method.
When multiplying such binomials, each term in the first binomial is multiplied by each term in the second binomial. However, the difference of squares formula allows an easier approach.

• The first term from each binomial, \(2m\), is squared: \((2m)^2 = 4m^2\).
• The second term from each binomial, \(5n\), is also squared: \((5n)^2 = 25n^2\).

By applying the formula directly, we encounter fewer steps and potential mistakes. Recognizing common patterns like this simplifies polynomial multiplication considerably.

Factoring Patterns

Factoring is a method used to simplify mathematical expressions by finding factors that multiply to form the original expression. One important pattern in factoring is the difference of squares. Our initial expression \((2m + 5n)(2m - 5n)\)exemplifies this pattern. In the difference of squares, we use the formula:

• \((a + b)(a - b) = a^2 - b^2\)

This efficient technique works because the middle terms, \(ab - ab\), cancel each other out, leaving us with \(a^2 - b^2\).

Such factoring patterns make solving complex problems easier. They allow for quick simplification and insight into the structure of polynomial expressions. Becoming familiar with these patterns is a practical tool in algebra, giving students a powerful shortcut when working through exercises.