Problem 141 - Introductory Algebra for College Students

Question

Factor completely. $$3 x^{2 n}-27 y^{2 n}$$

Step-by-Step Solution

Answer

The expression factors to \((\sqrt(3)x^n + 3\sqrt(3)y^n)(\sqrt(3)x^n - 3\sqrt(3)y^n)\)

Step 1: Identify the Difference of Squares

Rewrite the expression \(3x^{2n}-27y^{2n}\) as \((\sqrt(3)x^n)^2 - (\sqrt(27)y^n)^2\). Recognize this as a difference of squares with \(a = \sqrt(3)x^n\) and \(b = \sqrt(27)y^n\)

Step 2: Factor

Apply the formula \(a^2 - b^2 = (a+b)(a-b)\) with \(a = \sqrt(3)x^n, b =\sqrt(27)y^n\). This gives \((\sqrt(3)x^n + \sqrt(27)y^n)(\sqrt(3)x^n - \sqrt(27)y^n)\).

Step 3: Simplify

The expression \(\sqrt(27)\) can be simplified to \(\sqrt(9*3) = 3\sqrt(3)\). Replace \(\sqrt(27)\) with \(3\sqrt(3)\) to get \((\sqrt(3)x^n + 3\sqrt(3)y^n)(\sqrt(3)x^n - 3\sqrt(3)y^n)\)

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is fundamental in solving numerous mathematical problems. An algebraic expression is a mathematical phrase that can include numbers, variables (like x or y), and operation symbols such as +, −, ×, and ÷. It's an essential building block for all algebra, serving as a way to represent quantities and relationships between quantities symbolically.

For example, the expression

\(3x^{2n}-27y^{2n}\)

is an algebraic expression involving variables raised to a power, which in this case, is '2n'. The coefficient '3' and '27' are numerical factors that multiply with the variable terms. To make this expression more manageable or to simplify further calculations, we factor algebraic expressions. In our example, we aim to factor out the greatest common divisor and then use special factoring rules, such as the difference of squares, to simplify the expression.

Factoring Polynomials

Factoring polynomials is a critical skill that involves breaking down polynomials into simpler 'factorable' components or factors that, when multiplied together, will give back the original polynomial. This process is similar to finding 'ingredients' that come together to create the 'recipe' which is the polynomial itself.

One specific scenario in factoring is handling a difference of squares, which occurs when a polynomial can be written as a subtraction of two perfect squares, like \(a^2 - b^2\). The unique property of a difference of squares is that it can always be factored into \(a + b\) and \(a - b\). Applying this to our exercise, \(3x^{2n}-27y^{2n}\) is identified as a difference of squares and subsequently factored using the identity \(a^2 - b^2 = (a+b)(a-b)\), resulting in an easier-to-manage product of two binomials.

Exponent Rules

Exponent rules, also known as laws of exponents, are a set of rules that describe how to handle mathematical operations involving exponents. The three main operations we'll focus on are multiplication, division, and raising powers to powers. When you multiply terms with the same base, you add their exponents. When you divide them, you subtract the exponents. Finally, when raising an exponent to another power, you multiply the exponents.

These rules simplify expressions with exponents so that they are easier to work with. In the context of our original problem, knowing that \(27 = 3^3\) helps us simplify \(\sqrt(27)\) to \(3\sqrt(3)\), using our knowledge of exponents and radical expressions. This smooths the way to perfectly factor our algebraic expression using the difference of squares formula.