In Exercises 69–82, multiply using the rule for the product of the sum and difference of two terms.(x + 4)(x − 4)

Verified step by step guidance

1. Identify the expression as a product of the sum and difference of two terms: \((x + 4)(x - 4)\).
2. Recall the formula for the product of the sum and difference of two terms: \((a + b)(a - b) = a^2 - b^2\).
3. In this expression, identify \(a = x\) and \(b = 4\).
4. Apply the formula: substitute \(a\) and \(b\) into \(a^2 - b^2\) to get \(x^2 - 4^2\).
5. Simplify the expression \(x^2 - 4^2\) to complete the multiplication.

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 x and b is 4, allowing us to apply this rule directly.

Binomial Multiplication

Binomial multiplication involves multiplying two binomials, which are algebraic 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 these expressions is crucial for simplifying algebraic equations.

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 essential for recognizing patterns in polynomial expressions and simplifying them efficiently. In the context of the problem, applying this identity allows for a quick resolution of the multiplication without expanding the entire expression.