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Ch. R - Review of Basic Concepts
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. R - Review of Basic ConceptsProblem 96
Chapter 1, Problem 96

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

Verified step by step guidance
1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Solving Quadratic Equations Using The Quadratic Formula

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.
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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.
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Factor Using Special Product Formulas
Related Practice
Textbook Question

Add or subtract as indicated. 46.88 - 13.45

Textbook Question

Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. (k1/3)/(k2/3)(k-1)

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Textbook Question

Simplify each rational expression. Assume all variable expressions represent positive real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [(x2 +1)4(2x) - x2(4)(x2+1)3(2x)] / [(x2+1)8]

Textbook Question

Evaluate each expression. (-2/9 -1/4) - {-5/18 - (-1/2)}

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Textbook Question

Simplify each rational expression. Assume all variable expressions represent positive real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [(y2 +2)5(3y) - y3(6)(y2+2)4(3y)] / [(y2+2)7]

Textbook Question

Perform each division. See Examples 9 and 10. (4x3+9x2-10x-6)/(4x+1)