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

Verified step by step guidance

1. Recognize that the expression \( (x+y)^2 - (x-y)^2 \) is a difference of squares, which can be factored using the formula \( a^2 - b^2 = (a-b)(a+b) \). Here, let \( a = (x+y) \) and \( b = (x-y) \).
2. Apply the difference of squares formula: \( (x+y)^2 - (x-y)^2 = ((x+y) - (x-y))((x+y) + (x-y)) \).
3. Simplify the first factor: \( (x+y) - (x-y) = x + y - x + y = 2y \).
4. Simplify the second factor: \( (x+y) + (x-y) = x + y + x - y = 2x \).
5. Write the fully factored form as the product of the two simplified factors: \( 2y \times 2x = 4xy \). So the factored form is \( (2y)(2x) \) or simply \( 4xy \).

Key Concepts

Difference of Squares

The difference of squares is a factoring technique where an expression of the form a² - b² can be factored into (a - b)(a + b). Recognizing this pattern allows for quick simplification of expressions involving squared terms subtracted from each other.

Binomial Expansion

Binomial expansion involves expanding expressions like (x + y)² into x² + 2xy + y². Understanding how to expand binomials helps in identifying terms and simplifying expressions before factoring.

Factoring by Grouping

Factoring by grouping is a method where terms are grouped to find common factors, making it easier to factor complex expressions. This technique is useful when expressions do not immediately fit standard factoring formulas.