Problem 9 - Elementary and Intermediate Algebra

Question

Factor. \(x^{2}-16\)

Step-by-Step Solution

Answer

\((x - 4)(x + 4)\)

Step 1: Identify the Type of Expression

The expression provided is a binomial: \(x^2 - 16\). It appears to be a difference of squares. A difference of squares is an expression in the form \(a^2 - b^2\) where both terms are perfect squares.

Step 2: Recognize Perfect Squares

Identify the perfect squares in the expression. Here, \(x^2\) is the square of \(x\) and \(16\) is the square of \(4\). So, \(x^2 = (x)^2\) and \(16 = (4)^2\).

Step 3: Apply the Difference of Squares Formula

The formula for factoring a difference of squares is \(a^2 - b^2 = (a-b)(a+b)\). Here, \(a = x\) and \(b = 4\). Substitute these values into the formula to factor the expression.

Step 4: Factor the Expression

Using the difference of squares formula, \(x^2 - 16 = (x - 4)(x + 4)\). This is the fully factored form of the given expression.

Key Concepts

Factoring Binomials

Factoring binomials can initially seem tricky, but by recognizing patterns and applying formulas, it becomes straightforward. A binomial is an algebraic expression containing only two terms. The goal of factoring is to express it as a product of simpler expressions.
Often, binomials suitable for factoring, like the one in the problem, are "special binomials" such as a difference of squares or a perfect square trinomial (though a trinomial has three terms).

• Identifying special binomial cases can simplify factoring significantly.
• Look for two terms that can be the squares of two numbers.
• This will point you towards the difference of squares identity or other patterns.

For instance, in the expression given, which is a classic binomial, you can see it fits perfectly into a difference of squares, making it easier to factor.

Perfect Squares

Perfect squares are numbers or expressions that are the squares of an integer or another expression. Recognizing perfect squares is key in algebra, especially when factoring expressions like the one at hand.

• The number 16 in the expression is a perfect square since it equals \((4)^2\).
• Similarly, \(x^2\) is a perfect square since it equals \((x)^2\).

These perfect squares can simplify the process of applying the difference of squares formula.
When you recognize that both terms are perfect squares, you're halfway through factoring. The rest involves straightforward substitution into the difference of squares formula.

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and operations (like adding, subtracting, multiplying). They are foundational in algebra, forming the basis for equations and functions.

• A binomial is a type of algebraic expression. It has two terms, as seen in \(x^2 - 16\).
• The terms in an algebraic expression can often be rearranged or factored such that the expression appears in a simpler or more useful form.
• Expressions can be factored to solve equations, simplify expressions, or find intercepts among other common applications.

Understanding how to manipulate these expressions helps in uncovering hidden relationships between variables or numbers. When you factor \(x^2 - 16\), for instance, you reveal that \(x^2 - 16\) is equivalent to the product of two binomials, \((x - 4)(x + 4)\). This process eases solving equations where factored forms make operations like solving for roots much simpler.