Factor by any method. See Examples 1–7. (x+y)3−(x−$$y)3(x+y)^3$-(x-y)^3$$

Ch. R - Review of Basic Concepts

Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 109

1장, 문제 109

Factor by any method. See Examples 1–7. $(x + y)^{3} - (x - y)^{3}$

검증된 단계별 안내

Recognize that the expression is a difference of cubes: $ (x+y)^3 - (x-y)^3 $. This suggests using the formula for the difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, where $a = (x+y)$ and $b = (x-y)$.

Apply the difference of cubes formula: write the expression as $((x+y) - (x-y)) \times ((x+y)^2 + (x+y)(x-y) + (x-y)^2)$.

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

Expand and simplify each term inside the second factor: calculate $(x+y)^2$, $(x+y)(x-y)$, and $(x-y)^2$ separately.

Combine the expanded terms inside the second factor and simplify the expression to get the fully factored form.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Difference of Cubes

The difference of cubes formula states that a³ - b³ = (a - b)(a² + ab + b²). Recognizing expressions in the form of cubes allows factoring complex polynomials efficiently. In this problem, (x + y)³ and (x - y)³ are perfect cubes, enabling the use of this formula.

Binomial Expansion

Binomial expansion involves expanding expressions raised to a power, such as (x + y)³, using the binomial theorem or Pascal's triangle. Understanding the expanded form helps in verifying factorizations and simplifying expressions.

Factoring by Grouping

Factoring by grouping involves rearranging and grouping terms to find common factors. After applying the difference of cubes formula, grouping terms can simplify the expression further, making it easier to factor completely.

Factor by any method. See Examples 1–7. (x+y)3−(x−y)3(x+y)^3-(x-y)^3