Problem 44 - Beginning and Intermediate Algebra

Question

Factor completely. $$m^{2}-\frac{4}{25}$$

Step-by-Step Solution

Answer

The given expression can be factored completely as: \( (m + \frac{2}{5})(m - \frac{2}{5}) \) using the difference of squares formula.

Step 1: Identify the difference of squares formula

The difference of squares formula is: \( (a^2 - b^2) = (a+b)(a-b) \). We can use this formula to factor the given expression.

Step 2: Express the given expression as a difference of squares

Rewrite the given expression as: \( m^2 - \frac{4}{25} = m^2 - \left(\frac{2}{5}\right)^2 \)

Step 3: Apply the difference of squares formula

Using the formula, we can now substitute \(a = m\) and \(b = \frac{2}{5}\) in the formula: \( (m^2 - \left(\frac{2}{5}\right)^2) = (m + \frac{2}{5})(m - \frac{2}{5}) \)

Step 4: Write the final factored expression

The given expression is now factored completely as: \( (m + \frac{2}{5})(m - \frac{2}{5}) \)

Key Concepts

Difference of Squares

The difference of squares is a common pattern in algebra that simplifies expressions into a product of binomials. It is represented by the formula \( (a^2 - b^2) = (a + b)(a - b) \). This formula exploits the symmetry of squares to break them down into simpler factors. To apply the difference of squares successfully, both terms in the expression should be perfect squares themselves. By doing this, you can factor expressions more efficiently without extensive calculations. Recognizing this pattern is crucial in factoring polynomials quickly.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operators that come together to express a mathematical idea. They form the backbone of algebra and are used extensively to represent real-world problems mathematically. In the expression \( m^2 - \frac{4}{25} \), we see an excellent example of an algebraic expression involving subtraction of two terms. Such expressions can be straightforward, like monomials, or complex, like polynomials. Factoring, simplifying, and expanding algebraic expressions are essential techniques in mathematics that help in solving equations and inequalities.

Mathematics Education

Mathematics education aims to build foundational skills and problem-solving abilities. Understanding algebraic concepts, such as the difference of squares, plays a vital role in developing critical thinking skills. Educating students about various algebraic techniques, including factoring, helps them solve complex problems more effectively. Through engaging teaching methods and practice, students learn to apply these concepts in various contexts, enhancing their ability to reason and think logically. Mathematics education encourages exploring connections between different mathematical ideas, making the learning process more enriching.

Intermediate Algebra

Intermediate algebra acts as a bridge between basic arithmetic and advanced topics in mathematics. It introduces students to more advanced concepts like polynomial equations, factoring, and quadratic equations. Working with expressions like \( m^2 - \frac{4}{25} \) allows students to practice techniques such as the difference of squares. This level of algebra enhances students' ability to handle functions, manipulate algebraic expressions, and understand relationships between different mathematical operations. As students master intermediate algebra, they prepare for higher-level math courses, gaining confidence and proficiency in tackling various mathematical challenges.