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

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Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 113

1장, 문제 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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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