Ch. R - Review of Basic Concepts

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. R - Review of Basic Concepts

Problem 54

Chapter 1, Problem 54

Find each product. [(9r-s)+2][(9r-s)-2]

Verified step by step guidance

Recognize that the expression is in the form of a product of two binomials: \([(9r - s) + 2][(9r - s) - 2]\). This matches the pattern of a difference of squares: \((a + b)(a - b) = a^2 - b^2\).

Identify \(a\) and \(b\) in the expression: here, \(a = (9r - s)\) and \(b = 2\).

Apply the difference of squares formula: \((9r - s)^2 - 2^2\).

Expand the square of the binomial \((9r - s)^2\) using the formula \((x - y)^2 = x^2 - 2xy + y^2\), where \(x = 9r\) and \(y = s\).

Write the final expression as \(81r^2 - 18rs + s^2 - 4\), combining all parts without simplifying further.

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Key Concepts

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

Difference of Squares

The difference of squares is a special product formula expressed as (a + b)(a - b) = a² - b². It allows quick multiplication of two binomials that are conjugates, simplifying the expression by subtracting the square of the second term from the square of the first.

Binomial Expressions

A binomial is an algebraic expression containing two terms connected by addition or subtraction. Understanding how to manipulate binomials, including addition, subtraction, and multiplication, is essential for simplifying expressions and solving equations.

Exponentiation of Algebraic Terms

Exponentiation involves raising variables or expressions to a power, such as squaring a binomial term. Knowing how to correctly apply exponents to variables and constants is crucial for expanding and simplifying algebraic expressions.

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