Problem 25 - Elementary Algebra

Question

Factor completely. $$ x 2-y 2 $$

Step-by-Step Solution

Answer

\((x - y)(x + y)\)

Step 1: Identify the Expression

We have the expression \(x^2 - y^2\). This expression is a difference of squares.

Step 2: Recall the Difference of Squares Formula

The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\). This means that we can rewrite any expression of this form into a product of two binomials.

Step 3: Assign Values to Formula

In the expression \(x^2 - y^2\), assign \(a = x\) and \(b = y\). Then, apply the values to the difference of squares formula: \((x - y)(x + y)\).

Step 4: Complete the Factorization

Now apply the assigned values: \((x - y)(x + y)\). This is the completely factored form of \(x^2 - y^2\).

Key Concepts

Factoring

Factoring is a fundamental technique in algebra used to simplify expressions and solve equations. It involves breaking down complex expressions into simpler components, called factors. Think of it as unwrapping a present to see what's inside.Factoring helps to identify key properties of numbers and expressions, making them easier to handle. One of the most common forms of factoring is using special formulas, like the difference of squares. This specific formula comes in handy when you encounter expressions in the form of a square term subtracted from another square term, such as \(x^2 - y^2\).Remember, when you factor an expression, you're not changing its value. Instead, you're rewriting it in a way that's easier to work with. This is particularly useful when solving equations or simplifying expressions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators like addition or subtraction. They're like sentences in math language, with variables acting as the main characters.Variables, often represented by letters like \(x\) or \(y\), can take different values. The beauty of algebraic expressions lies in their ability to represent entire families of numbers with just one phrase. Consider \(x^2 - y^2\) from our original exercise. It features two variable terms and a subtraction operation, showcasing an algebraic expression's typical structure.Expressions can be manipulated using various operations, including factoring, to make them easier to work with. Once you understand the constituents of an expression, you can apply formulas and rules to transform and simplify them easily.

Binomials

A binomial is a specific type of algebraic expression that combines two terms. You can think of them as simple math sentences with two parts. Binomials are vital in various algebraic processes, including factoring and expanding expressions.In the context of factoring and the difference of squares, binomials play a critical role. Using the formula \(a^2 - b^2 = (a - b)(a + b)\), each side of the equation consists of binomials. Here, \((a - b)\) and \((a + b)\) are the two factors of the difference of squares.The beauty of binomials is their simplicity and the power they hold in algebraic manipulations. Recognizing binomials and understanding how to use them allows you to simplify and solve algebraic expressions efficiently.