Problem 37 - Elementary and Intermediate Algebra

Question

Factor. If a polynomial can't be factored, write "prime." $$ x^{2}-4 $$

Step-by-Step Solution

Answer

The polynomial factors to \((x - 2)(x + 2)\).

Step 1: Identify the Form

The polynomial given is \( x^2 - 4 \). This expression can be recognized as a difference of squares, which is structured in the form \( a^2 - b^2 \).

Step 2: Apply Difference of Squares Formula

The difference of squares formula states that \( a^2 - b^2 = (a-b)(a+b) \). For \( x^2 - 4 \), identify \( a = x \) and \( b = 2 \), since \( 4 = 2^2 \).

Step 3: Write the Factored Form Using the Formula

Substitute \( a \) and \( b \) into the formula: \( (x - 2)(x + 2) \). Thus, \( x^2 - 4 = (x - 2)(x + 2) \).

Key Concepts

Difference of Squares

When dealing with polynomials, the term "difference of squares" can frequently appear. It is all about recognizing a specific pattern: the expression must be structured as two squared terms subtracted from one another. For example, if we have an expression like \( x^2 - 4 \), this can be considered a difference of squares because \( x^2 \) and \( 4 \) (which is \( 2^2 \)) are both perfect squares.
Some important characteristics of a difference of squares include:

• There are always two terms in the expression.
• Each term should be a perfect square.
• It involves subtraction between the two squared terms.

This leads us to a neat trick: the difference of squares formula. It states that for any two numbers \( a \) and \( b \), \( a^2 - b^2 \) can be factored as \((a-b)(a+b)\). This formula helps simplify certain algebraic expressions efficiently.

Factoring Techniques

Factoring polynomials involves breaking down a polynomial into products of factors which, when multiplied together, give back the original polynomial. There are multiple techniques available for factoring, making it crucial to identify which one applies to a given problem.
Some common factoring techniques include:

• Factoring by grouping
• Factoring trinomials
• Using special formulas like the difference of squares

In our example of \( x^2 - 4 \), we use the difference of squares formula as the simplest way to factor it. Identifying \( a = x \) and \( b = 2 \), we apply the formula \( (x - 2)(x + 2) \). This transforms our expression into a product of simpler linear polynomials.
With practice, recognizing patterns and applying the correct technique becomes second nature.

Algebraic Expressions

Algebraic expressions are a core element of algebra, consisting of numbers, variables, and operations combined in a meaningful way. They form the building blocks for more complex equations and functions. An expression like \( x^2 - 4 \) involves operations (in this case, subtraction) and allows us to manipulate it for various purposes like factoring.
Understand that algebraic expressions:

• Contain coefficients, variables, and constants
• Can be simplified or factored
• Are foundational to solving equations

When working with these expressions, it’s important to become familiar with operations and transformations. Factoring is one such transformation that allows us to simplify an expression and solve equations efficiently. Recognizing patterns like the difference of squares within algebraic expressions provides a roadmap to achieve this simplification.