Problem 80 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The product is given by \(49x^2 - 9y^2\)

Step 1: Identify the elements

Identify \(a\) and \(b\) from the given product \((a+b)(a-b)\), which in this case is \((7x+3y)(7x-3y)\). So here, \(a=7x\) and \(b=3y\)

Step 2: Apply the Difference of Squares

Once you have identified \(a\) and \(b\), apply the Difference of Squares formula which is \(a^2 - b^2\). So substitute \(a = 7x\) and \(b = 3y\) into the formula to get \((7x)^2 - (3y)^2\)

Step 3: Calculate the square of 7x and 3y

At this step, calculate the square of two parts which make up the formula. \(a^2\) becomes \((7x)^2\) which equals to \(49x^2\) and \(b^2\) becomes \((3y)^2\) which equals to \(9y^2\)

Step 4: Subtract the terms

Subtract the squares of the two terms: \((49x^2 - 9y^2)\)

Key Concepts

Difference of Squares

The "Difference of Squares" is a crucial concept in algebra that can simplify seemingly complex expressions. Imagine you have two terms, \(a\) and \(b\). If you multiply such terms using the structure \((a+b)(a-b)\), you're dealing with a difference of squares. This is because it relates to the expression \(a^2 - b^2\).

Why is it called the "difference of squares"? Simply because, when you apply this rule, you end up subtracting one square term from another. Here’s why:

• The first term \(a+b\) multiplies with \(a-b\), and only the cross terms cancel each other out, leaving only \(a^2\) and \(b^2\).
• Hence, the result is \(a^2 - b^2\).

This approach simplifies calculations significantly, because instead of multiplying binomials step-by-step, you skip straight to the result in one swift operation. In the example given, \((7x+3y)(7x-3y)\), you'll quickly identify it as a difference of squares, thus simplifying the task to finding \((7x)^2 - (3y)^2\).

Algebraic Expressions

Algebraic expressions are built from numbers and variables, using arithmetic operations like addition, subtraction, multiplication, and division. They can represent real-life situations or abstract ideas mathematically. Understanding their structure is key to making sense of complex problems.

In the problem at hand, we have the expression \((7x+3y)(7x-3y)\). This expression involves two binomials, which are simplified using algebraic rules.
Here are some components of algebraic expressions, using our example as a reference:

• Terms: Individual parts of the expression separated by addition "\(+\)" or subtraction "\(-\)" signs. Here, \(7x\), \(3y\) are the parts when expression is broken down.
• Coefficients: The numerical multiplicative factor alongside variables. For \(7x\), the number 7 is a coefficient.
• Operators: Signs that indicate mathematical operations. "\(+\)" and "\(-\)" in this context.

Breaking down an algebraic expression into these parts helps in executing operations like multiplication, addition, subtraction and more, effectively.

Binomial Products

Binomial products stem from expressions with two terms, known as binomials. Often represented as \((a+b)\), these pairs hold a significant role in algebra. Multiplying such terms follows specific patterns, facilitating simplification.

The exercise provides a great example of this: multiplying \((7x+3y)\) and \((7x-3y)\). When you encounter two binomials, especially with the same elements but opposite signs like \((+\) and \(-\)), they form what's broadly known as "special products."

Here's how to approach binomial products:

• Distribute Each Term: Treat each term in the first binomial as a multiplier for every term in the second.
• For binomials of the form \((a+b)(a-b)\), the difference of squares formula makes the job easier, as cross-term products \((ab) - (ab)\) always cancel out.
• This leaves the squared terms, enabling the quick application of \(a^2-b^2\) formula.

Understanding binomial products simplifies expanding such expressions and serves as a foundational skill in algebraic manipulation.