Problem 125 - Introductory Algebra for College Students

Question

Factor each polynomial. $$x^{2 n}-25 y^{2 n}$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(x^{2n} - 25y^{2n}\) is \((x^n + 5y^n)(x^n - 5y^n)\).

Step 1: Identify the squares

First, identify the squares which make up the expression \(x^{2n} - 25y^{2n}\). The squares here are \(x^{2n}\) (let us consider this as \(a^2\)) and \(25y^{2n}\) (we will consider this as \(b^2\)). Note, each of \(a\) and \(b\) are positive.

Step 2: Apply the difference of squares formula

The formula states that the difference of two squares can be factored into the product of two binomials, \(a^2 - b^2 = (a+b)(a-b)\). Here, \(a\) corresponds to \(x^{n}\) and \(b\) is \(5y^{n}\).

Step 3: Substitute into formula

Plug \(a = x^n\) and \(b = 5y^n\) into the formula from step 2. This results in \(x^{2n} - 25y^{2n} = (x^n + 5y^n)(x^n - 5y^n)\).

Key Concepts

Difference of Squares

The concept of the difference of squares is a fundamental idea in algebra used in factoring polynomial expressions. It's called the "difference of squares" because it fundamentally deals with expressions that can be written as the subtraction (or difference) of two perfect squares. The difference of squares formula is given by: \[ a^2 - b^2 = (a+b)(a-b) \] This formula shows how an expression involving squares can be broken down into the product of two binomials.
When you identify expressions like \(x^{2n} - 25y^{2n}\), you can leverage this formula to simplify the problem into easier parts that are easier to work with. Such expressions are recognized by their structure: one term subtracted from another, and both terms being perfect squares. In our exercise, \(x^{2n}\) and \(25y^{2n}\) are both squares, allowing us to apply this concept easily.

Algebra

Algebra is a core branch of mathematics that deals with symbols and the rules for manipulating those symbols.
In the case of factoring polynomials like \(x^{2n} - 25y^{2n}\), algebra helps us abstract the problem and use known formulas to make complex expressions easier to solve. The main tool in this exercise is the difference of squares, which is one of many special formulas you learn in algebra. Understanding these foundational rules in algebra allows you to manipulate expressions and solve equations effectively.
Think of algebra as a language where the symbols and formulas act as grammar and vocabulary, helping you tell the story of how numbers relate to each other. In this particular exercise, knowing the formula allows you to transform and simplify expressions into a more manageable form.

Binomials

A binomial is an algebraic expression that contains exactly two terms. The term originates from "bi-" meaning two, and "nomial" meaning a part or term. In the difference of squares, the expression is transformed into a product of two binomials. In our example, the difference of squares formula turns \(x^{2n} - 25y^{2n}\) into \((x^n + 5y^n)(x^n - 5y^n)\). Each part of this product is a binomial with two distinct terms.
Understanding binomials is crucial for factoring in algebra because many problems boil down to expressing numbers or symbols as products of binomials or other expressions. Recognizing this form allows us to use simpler arithmetic or algebraic steps to arrive at a solution.
Mastering how to work with binomials gives you a powerful tool to tackle many problems in mathematics effectively.