Problem 49 - Elementary Algebra

Question

Factor completely. $$ x 2-64 y 2 $$

Step-by-Step Solution

Answer

The expression factors to \((x - 8y)(x + 8y)\).

Step 1: Identify the Expression

The expression given is \[ x^2 - 64y^2 \]We need to factor this expression completely.

Step 2: Recognize the Difference of Squares

Notice that the expression is in the form of \[ a^2 - b^2 \]where \( a = x \) and \( b = 8y \) since \( (8y)^2 = 64y^2 \).The difference of squares can be factored as \[ a^2 - b^2 = (a - b)(a + b) \]

Step 3: Apply the Difference of Squares Formula

Substitute \( x \) for \( a \) and \( 8y \) for \( b \) in the formula:\[(x - 8y)(x + 8y)\]

Step 4: Confirm that the Expression is Fully Factored

Check if any further factoring is possible. In this case, each binomial is in its simplest form with no common factors. Thus,\[(x - 8y)(x + 8y)\] is the completely factored expression.

Key Concepts

Difference of squares

The difference of squares is a valuable algebraic pattern used in factoring. It applies when you have an expression in the form \( a^2 - b^2 \). This pattern is one of the special cases of quadratic expressions that can be easily factored into two binomials. The formula used is \( a^2 - b^2 = (a - b)(a + b) \). Notice that it consists of two parts, each a perfect square, separated by a minus sign.

• \( a^2 \) represents the square of a, which is the first term.
• \( b^2 \) represents the square of b, which is the second term.

For the expression \( x^2 - 64y^2 \), we recognize \( x^2 \) as \( a^2 \) and \( (8y)^2 \) as \( b^2 \), identifying a as \( x \) and b as \( 8y \). This structure perfectly fits the difference of squares pattern and can be decomposed into \((x - 8y)(x + 8y)\). This factored form cancels out the squares through the difference pattern, simplifying the expression and allowing for easier computation in further algebraic manipulations.

Binomials

Binomials are specific types of algebraic expressions that contain exactly two terms. These terms are usually connected by addition or subtraction. Binomials form the building blocks for many polynomials, which makes them essential to understand in algebra. For example, in our step-by-step solution, we deal with the binomials \( (x - 8y) \) and \( (x + 8y) \). Both expressions consist of two terms

Here's what you need to know about binomials:

• Each binomial has a simple yet versatile structure, comprising a sum or difference of two elements, such as \( a \) and \( b \).
• They can be further manipulated through operations like addition, subtraction, and multiplication, or factored further if they contain common factors.

Recognizing binomials and exploiting their properties allow for streamlined factoring of more complex algebraic expressions. In the case of a difference of squares, the binomial factors provide a condensed, simplified form of the original expression, ready to be used in further calculations without complicating the process unnecessarily.

Algebraic expressions

Algebraic expressions form the language of algebra. They include variables, constants, and operation symbols. Understanding how to handle algebraic expressions is critical for solving equations, factoring, and simplifying calculations. Expressions can vary from simple to complex, depending on the number and type of operations involved.

Let's break down the expression \( x^2 - 64y^2 \):

• \( x^2 \) and \( 64y^2 \) are termed as quantifiable parts of the expression.
• The subtraction operator indicates how these parts are combined.

Factoring simplifies algebraic expressions by breaking them into components that are easier to work with. Using the specific method of the difference of squares, as demonstrated in this solution, highlights one of these strategies. Through careful recognition of patterns within the expressions, such as perfect squares and simple binomials, the complexity diminishes. This leads to more straightforward problem-solving and calculations in algebraic contexts, making the study of these expressions both practical and empowering.