Factor each polynomial. See Examples 5 and 6. (p-2q)2-100

Ch. R - Review of Basic Concepts

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Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 66

1장, 문제 66

Factor each polynomial. See Examples 5 and 6. (p-2q)²-100

검증된 단계별 안내

Recognize that the given expression \( (p-2q)^2 - 100 \) is a difference of squares, since it can be written as \( (p-2q)^2 - 10^2 \).

Recall the difference of squares formula: \( a^2 - b^2 = (a - b)(a + b) \). Here, \( a = p - 2q \) and \( b = 10 \).

Apply the formula by substituting \( a \) and \( b \) into it: \( (p - 2q - 10)(p - 2q + 10) \).

Write the factored form clearly as \( (p - 2q - 10)(p - 2q + 10) \).

Verify by expanding the factors to ensure they multiply back to the original expression.

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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 used when an expression is in the form a² - b². It factors into (a - b)(a + b). Recognizing this pattern helps simplify expressions like (p - 2q)² - 100 by identifying a = (p - 2q) and b = 10.

Binomial Squares

A binomial square is an expression of the form (x ± y)², which expands to x² ± 2xy + y². Understanding how to expand and recognize binomial squares is essential for factoring or simplifying expressions involving squared binomials, such as (p - 2q)².

Polynomial Factoring

Polynomial factoring involves rewriting a polynomial as a product of simpler polynomials. Mastery of various factoring methods, including factoring by grouping, special products, and difference of squares, is crucial for breaking down complex expressions into factors.