In Exercises 15–58, find each product. (1−y5)(1+y5)

Ch. P - Fundamental Concepts of Algebra

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Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

문제 39

1장, 문제 39

In Exercises 15–58, find each product. (1−y⁵)(1+y⁵)

검증된 단계별 안내

Recognize that the given expression (1−y^5)(1+y^5) is a product of two binomials in the form (a−b)(a+b). This is a difference of squares formula.

Recall the difference of squares formula: (a−b)(a+b) = a^2 − b^2. Here, a = 1 and b = y^5.

Substitute the values of a and b into the formula: a^2 − b^2 becomes 1^2 − (y^5)^2.

Simplify each term: 1^2 simplifies to 1, and (y^5)^2 simplifies to y^(5×2) = y^10.

Combine the simplified terms to get the final expression: 1 − y^10.

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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 fundamental algebraic identity that states that for any two terms a and b, the expression (a - b)(a + b) equals a² - b². This identity is crucial for simplifying expressions that are structured as a product of a sum and a difference, allowing for easier calculations and factorizations.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors, which can simplify expressions and solve equations. In the context of the given expression, recognizing the structure as a difference of squares allows for straightforward factoring, leading to a more manageable form.

Algebraic Manipulation

Algebraic manipulation refers to the techniques used to rearrange and simplify algebraic expressions. This includes applying identities, combining like terms, and distributing products. Mastery of these techniques is essential for effectively solving algebraic problems and understanding the relationships between different algebraic forms.