Problem 42 - Algebra 1: Concepts and Skills

Question

Factor the expression. $$ m^{2}-25 $$

Step-by-Step Solution

Answer

The factored form of the expression \(m^2 - 25\) is \((m+5)(m-5)\).

Step 1: Recognize the structure

The first step is to recognize that the expression \(m^2 - 25\) is in the form of \(a^2 - b^2\), which indicates a difference of squares. Here \(a = m\) and \(b = 5\) because \(m^2 = (m)^2\) and \(25 = (5)^2\).

Step 2: Apply the formula

By applying the difference of squares formula, which states that \(a^2 - b^2 = (a+b)(a-b)\), the given equation becomes \((m+5)(m-5)\).

Step 3: Writing the final answer

The final factored expression is \((m+5)(m-5)\), as found using the difference of squares formula.

Key Concepts

Difference of Squares

The difference of squares is a common technique in algebra used to simplify expressions, specifically quadratic expressions like the one given: \( m^2 - 25 \). This method relies on the identity \( a^2 - b^2 = (a+b)(a-b) \). Recognizing the expression as a difference of squares is crucial because it allows us to break them down into simpler terms. In our example, \( m^2 \) is the square of \( m \) and \( 25 \) is the square of \( 5 \). Thus, the expression can be rewritten using the formula to become \( (m+5)(m-5) \). This process not only simplifies the expression but also helps in solving equations where such factored forms are needed.

Algebraic Structures

Understanding algebraic structures is fundamental when working with expressions. In the context of the given expression \( m^2 - 25 \), recognizing this as a binomial expression is key. Binomials are expressions with two terms, like our example which can be broken down using algebraic identities.

Algebraic structures like polynomials and binomials serve as building blocks for more complex expressions. The difference of squares formula used here is an algebraic structure itself, simplifying how we handle expressions and solve equations.

Recognizing these structures not only enhances problem-solving skills but also deepens comprehension, making seemingly complex problems more approachable. By focusing on the familiar pattern of the difference of squares, we efficiently factor and resolve the expression into its simpler components \( (m+5) \) and \( (m-5) \).

Factoring Process

The factoring process involves breaking down equations or expressions into simpler, easily manageable parts, often factors or products of simpler expressions. The example given, \( m^2 - 25 \), demonstrates this well. Factoring this involves recognizing the expression as a difference of squares and applying the corresponding formula.

Here are the steps:

• Identify the expression format: Compare the given expression with classic forms, such as \( a^2 - b^2 \).
• Determine components: Here, \( m^2 \) and \( 25 \) are recognized as squares of \( m \) and \( 5 \) respectively.
• Apply the formula: Use \( (a+b)(a-b) \) to factor the expression.
• Rewrite the expression: Conclude with the factored form \( (m+5)(m-5) \).

The factoring process transforms a more complex expression into a product of simpler expressions, making analysis and solution finding much more straightforward. Understanding this process is invaluable for students tackling various algebraic problems.