Problem 58 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. $$(7 m+2 n)(7 m-2 n)$$

Step-by-Step Solution

Answer

The result is \(49m^2 - 4n^2\).

Step 1: Recognize the Expression Type

The given expression \((7m + 2n)(7m - 2n)\) is a difference of squares formula. It takes the form of \((a + b)(a - b) = a^2 - b^2\).

Step 2: Identify 'a' and 'b'

Here, identify \(a = 7m\) and \(b = 2n\) corresponding to the formula \((a + b)(a - b)\).

Step 3: Apply the Difference of Squares Formula

Using the formula \(a^2 - b^2\), substitute \(a = 7m\) and \(b = 2n\): \[(7m)^2 - (2n)^2\].

Step 4: Simplify the Expression

Calculate \( (7m)^2 = 49m^2 \) and \( (2n)^2 = 4n^2 \), then write the result: \[49m^2 - 4n^2\].

Step 5: Write the Final Answer

The simplified result of the given operation is \(49m^2 - 4n^2\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators. They lay the groundwork for much of algebra and help describe and solve problems involving unknown quantities. In an expression like \(7m + 2n\), \(7\) and \(2\) are constants, while \(m\) and \(n\) are variables.
The operators in algebraic expressions include addition, subtraction, multiplication, and division. When working with algebraic expressions, you can perform special operations and apply various rules to simplify or manipulate them. Such operations often help in factoring, solving equations, and even calculus. This problem emphasizes recognizing patterns in algebraic expressions to simplify expressions more efficiently.

Factoring Techniques: Difference of Squares

Factoring is a crucial technique in algebra that involves breaking down an expression into a product of simpler terms. This can make solving equations and simplifying expressions much easier. One commonly used factoring technique is the "Difference of Squares." This method applies to any expression of the form \((a+b)(a-b)\).

• Firstly, identify the values for \(a\) and \(b\) in the expression \((7m + 2n)(7m - 2n)\).
• Observe that this structure matches \((a+b)(a-b) = a^2 - b^2\), a special product.
• This tells us the expression can be simplified to \((7m)^2 - (2n)^2\).

Using the difference of squares formula is an efficient way to factor certain expressions quickly. With practice, it becomes easy to spot these patterns, saving time on lengthy calculations.

Exploring Polynomials and Their Products

Polynomials are algebraic expressions made up of variables and coefficients, connected by addition, subtraction, and multiplication. They are an essential part of algebra and serve as the building blocks for more complex mathematical concepts. A polynomial can vary in terms of its degree—the highest power of the variable within the expression.

• The given exercise involves a product of two binomials, \((7m + 2n)\) and \((7m - 2n)\), which result in a polynomial of degree 2 when expanded and simplified.
• Once you recognize the expressions as a difference of squares, you can directly apply the formula, yielding \(49m^2 - 4n^2\).
• This result represents a simplified polynomial of degree 2, composed of the terms \(49m^2\) and \(-4n^2\).

Understanding how to form and manipulate polynomials is fundamental to mastering algebra. They come in various forms and degrees, but knowing rules like the difference of squares helps in swift simplification and understanding.