Problem 74 - Algebra 2

Question

Factor each polynomial. $$ x^{2}-16 $$

Step-by-Step Solution

Answer

The polynomial \(x^2 - 16\) factors to \((x - 4)(x + 4)\).

Step 1: Recognize the Structure

The polynomial \(x^2 - 16\) is a difference of squares. The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\). Identify \(a\) and \(b\) for this expression.

Step 2: Identify Values of 'a' and 'b'

In the expression \(x^2 - 16\), the term \(x^2\) represents \(a^2\) and \(16\) is \(b^2\). Thus, we have \(a = x\) and \(b = 4\).

Step 3: Apply the Difference of Squares Formula

Plug the values of \(a\) and \(b\) into the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\). This results in \((x - 4)(x + 4)\).

Step 4: Verify the Factored Form

Expand \((x - 4)(x + 4)\) to check if it equals the original polynomial. \((x - 4)(x + 4) = x^2 + 4x - 4x - 16 = x^2 - 16\). The expanded form matches the original polynomial, confirming the factors.

Key Concepts

Difference of Squares

In algebra, the difference of squares is a specific pattern that can make the factoring process much simpler. It applies when you have two square terms separated by a subtraction sign. The general form is written as \(a^2 - b^2\), which can be factored into two binomials: \((a - b)(a + b)\). Recognizing this structure allows us to factor expressions quickly and accurately.
For instance, in the expression \(x^2 - 16\), we can observe it follows the difference of squares pattern. Here, \(x^2\) and \(16\) are both perfect squares, where \(x^2\) is \(a^2\) and \(16\) is \(b^2\) with \(b = 4\). When applied to the formula, it factors as \((x - 4)(x + 4)\), showcasing its simplicity and efficiency.

Factoring Techniques

Mastering various factoring techniques is crucial in simplifying algebraic expressions and solving polynomial equations. One of the primary techniques involves recognizing patterns, such as the difference of squares. This pattern not only saves time but also reduces complexity during factorization.
When facing a polynomial like \(x^2 - 16\), we can swiftly identify it as a difference of squares because it fits the formula \(a^2 - b^2\). Once identified, this pattern informs us to split the polynomial using the two binomial factors \((a - b)(a + b)\). In our example, \(x^2 - 16\) becomes \((x - 4)(x + 4)\).
These strategic techniques transform seemingly complex algebraic tasks into manageable steps, promoting accuracy and speed in math work.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They are fundamental in algebra and come in numerous forms, such as monomials, binomials, and polynomials.
A key challenge is simplifying or factoring these expressions, a step that often requires identifying specific patterns. For example, the polynomial \(x^2 - 16\) is an algebraic expression that can be simplified by recognizing it as a difference of squares.
Simplifying algebraic expressions through factoring not only aids in solving equations but also reveals deeper relationships within expressions. By breaking down \(x^2 - 16\) into \((x - 4)(x + 4)\), we convert it into a simplified form that is easier to analyze and interpret.