Problem 80 - Algebra and Trigonometry

Question

In Exercises 67–82, find each product. $$ (7 x+3 y)(7 x-3 y) $$

Step-by-Step Solution

Answer

\(49x^2 - 9y^2\)

Step 1: Recognize the Pattern

From the expression given, \( (7x+3y)(7x-3y)\), it can be observed that it follows the format of \((a+b)(a-b)\) where \(a=7x\) and \(b=3y\).

Step 2: Apply the Formula

In \((a+b)(a-b)\), it simplifies to \(a^2 - b^2\). So apply this rule to the given expression. This will simplify the equation to \( (7x)^2 - (3y)^2\).

Step 3: Simplify the Expression

Next, simplify the equation by squaring the individual terms. When \(7x\) is squared, it becomes \(49x^2\). Similarly, when \(3y\) is squared, it becomes \(9y^2\). This gives \(49x^2 - 9y^2\) as the simplified expression.

Key Concepts

Binomial Multiplication

Binomial multiplication refers to the process of multiplying two binomial expressions together. A binomial is a polynomial with two terms, such as

• \((a + b)\)
• \((a - b)\)

In our exercise, we have a slightly more complex example with the terms \((7x + 3y)\) and \((7x - 3y)\). These are typical binomials where both consist of two terms each. Binomial multiplication utilizes specific algebraic patterns, such as \((a + b)(a - b)\), to simplify computations. This specific pattern is known as the difference of squares because it results in a difference (subtraction) between square terms when expanded or simplified.

Algebraic Patterns

Recognizing algebraic patterns can greatly simplify the process of multiplying binomials. One of the most common patterns in algebra is \((a+b)(a-b) = a^2 - b^2\). This is known as the difference of squares, and it happens when two terms are conjugates of each other.

• For instance, our exercise demonstrates this pattern with \((7x + 3y)\) and \((7x - 3y)\). Recognizing that this follows the \((a + b)(a - b)\) pattern, we can directly use the formula which simplifies to \(a^2 - b^2\).
• By recognizing these patterns, we can turn seemingly complex multiplications into straightforward algebraic operations.
  

This saves time and reduces errors especially when dealing with larger expressions. Hence, identifying and using algebraic patterns effectively is crucial in mastering algebra.

Simplifying Expressions

Simplifying expressions is the process of making an expression as easy to work with as possible. This involves eliminating unnecessary terms and operations. In the exercise, once we've identified the \((a + b)(a - b)\) pattern in \((7x + 3y)(7x - 3y)\), the next step is about simplifying it using the difference of squares:

• First, calculate \((7x)^2\), which gives \(49x^2\).
• Next, calculate \((3y)^2\), resulting in \(9y^2\).
• Finally, subtract these results to get the simplified expression \(49x^2 - 9y^2\).

Using properties like squaring and subtraction effectively clarifies the expression. It's essential for ensuring that the final expression is both accurate and straightforward to use in further algebraic operations or problem-solving scenarios.