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

Ch. R - Review of Basic Concepts

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. R - Review of Basic Concepts

Problem 113

Chapter 1, Problem 113

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

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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) $.

Apply the difference of squares formula: $ (x+y)^2 - (x-y)^2 = ((x+y) - (x-y))((x+y) + (x-y)) $.

Simplify the first factor: $ (x+y) - (x-y) = x + y - x + y = 2y $.

Simplify the second factor: $ (x+y) + (x-y) = x + y + x - y = 2x $.

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 $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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