Factor each polynomial. See Examples 5 and 6. (a+b)2-16

Ch. R - Review of Basic Concepts

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Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 65

1장, 문제 65

Factor each polynomial. See Examples 5 and 6. (a+b)²-16

검증된 단계별 안내

Recognize that the given expression $ (a+b)^2 - 16 $ is a difference of squares, since $16$ can be written as \$4^2$.

Recall the difference of squares formula: $x^2 - y^2 = (x - y)(x + y)$.

Identify $x$ as $(a+b)$ and $y$ as $4$ in the expression $ (a+b)^2 - 4^2 $.

Apply the difference of squares formula: $((a+b) - 4)((a+b) + 4)$.

Simplify each binomial to get the factored form: $(a + b - 4)(a + b + 4)$.

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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 (a+b)² - 16, where 16 is a perfect square.

Expanding and Recognizing Perfect Squares

Understanding how to expand and recognize perfect square trinomials, such as (a+b)² = a² + 2ab + b², is essential. This knowledge helps identify when an expression can be rewritten or factored using special formulas.

Polynomial Factoring Techniques

Factoring polynomials involves rewriting them as products of simpler polynomials. Techniques include factoring out the greatest common factor, grouping, and applying special formulas like difference of squares or perfect square trinomials.