Factor by any method. See Examples 1–7. (2y−1)2−4(2y−$$1)+4(2y-1)^2-4(2y-1)+4$$

Ch. R - Review of Basic Concepts

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Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 96

1장, 문제 96

Factor by any method. See Examples 1–7. $(2 y - 1)^{2} - 4 (2 y - 1) + 4$

검증된 단계별 안내

Recognize that the expression is in terms of the binomial $ (2y - 1) $. Let $ u = (2y - 1) $ to simplify the expression.

Rewrite the expression using $ u $: $ u^2 - 4u + 4 $.

Notice that $ u^2 - 4u + 4 $ is a quadratic trinomial. Check if it is a perfect square trinomial by comparing it to the form $ a^2 - 2ab + b^2 = (a - b)^2 $.

Since $ u^2 - 4u + 4 = (u - 2)^2 $, rewrite the expression back in terms of $ y $: $ ((2y - 1) - 2)^2 $.

Simplify inside the parentheses: $ (2y - 1 - 2)^2 = (2y - 3)^2 $. This is the factored form of the original expression.

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주요 개념

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Recognizing Quadratic Expressions

A quadratic expression is a polynomial of degree two, often in the form ax² + bx + c. In this problem, the expression is written in terms of a binomial (2y - 1), which can be treated as a single variable to simplify factoring.

Substitution Method for Factoring

Substitution involves replacing a complex expression with a single variable to make factoring easier. Here, letting u = (2y - 1) transforms the expression into u² - 4u + 4, a standard quadratic that can be factored using familiar methods.

Factoring Perfect Square Trinomials

A perfect square trinomial is of the form a² - 2ab + b², which factors into (a - b)². Recognizing that u² - 4u + 4 fits this pattern allows quick factoring into (u - 2)², which can then be rewritten in terms of y.

Factor by any method. See Examples 1–7. (2y−1)2−4(2y−1)+4(2y-1)^2-4(2y-1)+4