Factor by any method. See Examples 1–7. (x+y)^2-(x-y)^2

Verified step by step guidance

1. Recognize the expression as a difference of squares: \((x+y)^2 - (x-y)^2\).
2. Recall the difference of squares formula: \(a^2 - b^2 = (a-b)(a+b)\).
3. Identify \(a = (x+y)\) and \(b = (x-y)\) in the expression.
4. Apply the difference of squares formula: \((x+y) - (x-y)\) and \((x+y) + (x-y)\).
5. Simplify each part: \((x+y) - (x-y) = 2y\) and \((x+y) + (x-y) = 2x\).

Key Concepts

Difference of Squares

The difference of squares is a fundamental algebraic identity that states a² - b² = (a - b)(a + b). This concept is crucial for factoring expressions that can be represented as the difference between two squared terms. In the given expression, (x+y)² and (x-y)² are both perfect squares, allowing us to apply this identity effectively.

Factoring Techniques

Factoring techniques involve rewriting an expression as a product of its factors. Common methods include grouping, using identities like the difference of squares, and recognizing patterns in polynomials. Understanding these techniques is essential for simplifying expressions and solving equations, as they allow for easier manipulation of algebraic terms.

Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power, such as (a + b)² = a² + 2ab + b². This concept is important for recognizing how to manipulate and simplify expressions involving binomials. In the context of the question, recognizing the squared binomials helps in applying the difference of squares effectively.