Find each product. (7x+4y)(7x-4y)

Verified step by step guidance

1. Recognize that the given expression (7x+4y)(7x-4y) is a product of two binomials in the form (a+b)(a-b), which is a difference of squares formula.
2. Recall the difference of squares formula: (a+b)(a-b) = a² - b².
3. Identify the terms in the binomials: a = 7x and b = 4y.
4. Substitute the identified terms into the formula: a² - b² becomes (7x)² - (4y)².
5. Simplify each squared term: (7x)² = 49x² and (4y)² = 16y², so the expression simplifies to 49x² - 16y².

Key Concepts

Binomial Multiplication

Binomial multiplication involves multiplying two binomials, which are algebraic expressions containing two terms. The process typically uses the distributive property, often summarized by the acronym FOIL (First, Outside, Inside, Last) to ensure all combinations of terms are multiplied. For example, in the expression (a + b)(c + d), each term in the first binomial is multiplied by each term in the second.

Difference of Squares

The difference of squares is a specific algebraic identity that states that the product of two conjugates, such as (a + b)(a - b), equals a² - b². This identity simplifies the multiplication of binomials where one is the negative of the other, allowing for quick calculations and simplifications. In the given expression, (7x + 4y)(7x - 4y) can be recognized as a difference of squares.

Algebraic Simplification

Algebraic simplification is the process of reducing an expression to its simplest form by combining like terms and applying algebraic identities. This is crucial in making complex expressions easier to work with and understand. After applying the difference of squares to the expression (7x + 4y)(7x - 4y), the result can be simplified to 49x² - 16y², showcasing the importance of this concept.