In Exercises 67–82, find each product. (7xy2−10y)(7xy2+10y)

Ch. P - Fundamental Concepts of Algebra

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Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

문제 81

1장, 문제 81

In Exercises 67–82, find each product. (7xy²−10y)(7xy²+10y)

검증된 단계별 안내

Step 1: Recognize that the given expression is a product of two binomials, (7xy² - 10y) and (7xy² + 10y). This is a special case of the difference of squares formula: (a - b)(a + b) = a² - b².

Step 2: Identify the terms 'a' and 'b' in the binomials. Here, 'a' is 7xy² and 'b' is 10y.

Step 3: Apply the difference of squares formula. Substitute 'a' and 'b' into the formula: a² - b² = (7xy²)² - (10y)².

Step 4: Simplify each squared term. For (7xy²)², square both the coefficient and the variables: (7²)(x²)(y⁴). For (10y)², square the coefficient and the variable: (10²)(y²).

Step 5: Write the simplified expression. Combine the results from Step 4 to express the product as: 49x²y⁴ - 100y².

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주요 개념

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

Factoring and the Difference of Squares

The expression given is in the form of a difference of squares, which follows the identity a^2 - b^2 = (a - b)(a + b). In this case, 7xy^2 is 'a' and 10y is 'b'. Recognizing this pattern allows for efficient multiplication and simplification of the expression.

Distributive Property

The distributive property states that a(b + c) = ab + ac. This property is essential for expanding polynomials, as it allows us to multiply each term in one polynomial by each term in the other. Understanding how to apply this property is crucial for correctly finding the product of the two binomials.

Combining Like Terms

After applying the distributive property, the resulting expression may contain like terms, which are terms that have the same variable factors raised to the same powers. Combining like terms involves adding or subtracting these terms to simplify the expression, leading to a more concise and manageable result.