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Chapter 6: Problem 141

In Exercises \(137-141,\) factor completely. $$3 x^{2 n}-27 y^{2 n}$$

Short Answer

Expert verified
The completely factorized form of \(3x^{2n} - 27y^{2n}\) is \(3(x^n - 3y^n)(x^n + 3y^n)\)

Step by step solution

01

Identify Common Factors

Look for common factors in each term of the expression. In this case, the common factor is 3: \(3 x^{2n}-27 y^{2n} = 3(x^{2n}-9y^{2n})\).
02

Apply the Difference of Two Squares

Knowing that \( x^{2n} = (x^n)^2 \) and \( 9y^{2n} = (3y^n)^2 \), the expression can be rewritten and factored using the formula of difference of two squares: \(3(x^{2n}-9y^{2n}) = 3((x^n)^2 - (3y^n)^2) = 3(x^n - 3y^n)(x^n + 3y^n)\).
03

Final Factorized Form

After taking common factors and applying the formula, the expression is now factored completely: \( 3(x^{2n} - 27y^{2n}) = 3(x^n - 3y^n)(x^n + 3y^n) \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Two Squares
The difference of two squares is a powerful tool in algebra that refers to an expression of the form \(a^2 - b^2\) and can be factored into \(a - b)(a + b)\). This is an essential concept because it simplifies seemingly complex problems into more manageable ones. For instance, when you encounter an algebraic expression such as \(x^{2n} - 9y^{2n}\), you can recognize it as the difference of two squares if you rewrite it as \( (x^n)^2 - (3y^n)^2 \), which then factors into \( (x^n - 3y^n)(x^n + 3y^n) \). This foundational concept in intermediate algebra not only streamlines calculations, but also enhances the understanding of more advanced mathematical topics.
Common Factor
Identifying a common factor in algebraic expressions is like finding a shortcut to solving a complex problem. A common factor is a term that is present in each term of an expression. For example, take the expression \(3x^{2n} - 27y^{2n}\), where the number 3 is a common factor. To factor this expression, you start by extracting the common factor: \(3(x^{2n} - 9y^{2n})\). Here, by pulling out the common factor, the problem is reduced to a simpler form that is easier to handle. Recognizing common factors is an essential skill for students learning intermediate algebra, as it can greatly simplify the process of factoring polynomials and solving equations.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations (like addition and multiplication) that together represent a specific quantity. For example, the polynomial \(3x^{2n} - 27y^{2n}\) is an algebraic expression with terms that are made up of base variables (\(x\) and \(y\)), exponents (\(2n\)), and coefficients (3 and -27). When working with algebraic expressions, understanding properties such as the distributive law, and the ability to recognize patterns, like the difference of two squares, are vital. Creating a solid foundation in working with algebraic expressions sets the stage for mastering more complex concepts in algebra.
Intermediate Algebra
Intermediate algebra is the branch of mathematics that bridges the gap between basic algebraic concepts learned in early education and more advanced topics. It includes the study of polynomial expressions, factoring techniques, equations, and functions. Understanding the methods of factoring, such as recognizing a difference of two squares or extracting a common factor, is integral to intermediate algebra. This level of algebra prepares students with the skills necessary to approach and solve a wide range of mathematical problems and serves as a stepping stone towards more abstract and higher-level math studies.

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Most popular questions from this chapter

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