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

Verified step by step guidance

1. Identify the expression as a product of the sum and difference of two terms: \((x + 5)(x - 5)\).
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 \(b = 5\).
4. Apply the formula: substitute \(a\) and \(b\) into \(a^2 - b^2\) to get \(x^2 - 5^2\).
5. Simplify the expression by calculating \(5^2\) to complete the expression \(x^2 - 25\).

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. Understanding this concept allows for quicker calculations and a clearer grasp of polynomial identities.

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. Mastery of this concept is essential for simplifying expressions and solving equations in algebra.

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 in algebra as it provides a method for factoring quadratic expressions and solving equations. Recognizing this pattern helps in simplifying complex expressions and understanding polynomial behavior.