In Exercises 69–82, multiply using the rule for the product of the sum and difference of two terms.(5x + 3)(5x − 3)

Verified step by step guidance

1. Identify the expression as a product of the sum and difference of two terms: \((a + b)(a - b)\).
2. Recognize that this expression can be simplified using the difference of squares formula: \((a + b)(a - b) = a^2 - b^2\).
3. In the given expression \((5x + 3)(5x - 3)\), identify \(a = 5x\) and \(b = 3\).
4. Apply the difference of squares formula: \((5x)^2 - (3)^2\).
5. Simplify the expression by calculating \((5x)^2\) and \(3^2\) separately, then subtract the results.

Key Concepts

Product of Sum and Difference

The product of the sum and difference of two terms follows the formula (a + b)(a - b) = a² - b². This identity simplifies the multiplication of binomials by eliminating the middle terms, resulting in a difference of squares. In the given expression, a is 5x and b is 3, allowing for straightforward application of this rule.

Binomial Multiplication

Binomial multiplication involves multiplying two binomials, which are expressions containing two terms. The process can be executed using the distributive property or special product formulas, such as the one for the product of the sum and difference. Understanding how to manipulate binomials is essential for simplifying algebraic expressions efficiently.

Difference of Squares

The difference of squares is a specific algebraic identity that states a² - b² can be factored into (a + b)(a - b). This concept is crucial when simplifying expressions that fit this form, as it allows for quick calculations and insights into the properties of quadratic expressions. Recognizing this pattern is key to solving problems involving products of binomials.