Problem 57 - Elementary and Intermediate Algebra

Question

Factor. $$ 8 x^{2}-32 y^{2} $$

Step-by-Step Solution

Answer

The expression factors to \(8(x + 2y)(x - 2y)\).

Step 1: Identify a common factor

First, observe the expression \(8x^2 - 32y^2\). Both terms share a common factor of 8, the greatest common factor. Start by factoring out 8 from the expression: \[8(x^2 - 4y^2)\].

Step 2: Recognize the difference of squares

The expression inside the parentheses \(x^2 - 4y^2\) is a difference of squares. Recall that the difference of squares formula is \(a^2 - b^2 = (a + b)(a - b)\).

Step 3: Apply the difference of squares formula

Now we apply the difference of squares formula to \(x^2 - (2y)^2\) where \(a = x\) and \(b = 2y\). Thus, \(x^2 - 4y^2\) becomes \((x + 2y)(x - 2y)\).

Step 4: Combine the factors

Bring back the factor of 8 that we extracted in the first step. The final factored form of \(8x^2 - 32y^2\) is \[8(x + 2y)(x - 2y)\].

Key Concepts

Greatest Common Factor

When simplifying algebraic expressions, identifying the greatest common factor (GCF) is often the first step. The GCF is the largest factor that divides all terms in an expression without leaving any remainder. Finding the GCF makes it easier to simplify or factor expressions.
For the expression given, \(8x^2 - 32y^2\), we notice both terms, \(8x^2\) and \(32y^2\), have a common factor. The number 8 is the greatest factor that can divide both 8 and 32. By factoring out the GCF, the expression simplifies, leaving us with fewer and simpler factors to work with.
Here's how you do it:

• Find the GCF for the coefficients (numbers in front of the variables), which here is 8.
• Factor out the GCF, rewriting the expression as \(8(x^2 - 4y^2)\).

This step is crucial because it sets up the expression for further simplification or solving.

Difference of Squares

The 'difference of squares' is a special algebraic pattern. It occurs when you have an expression of the form \(a^2 - b^2\). The beauty of this form is that it can be factored easily into \((a + b)(a - b)\). This pattern is extremely common and useful in algebra because it simplifies complex expressions quickly.
In our exercise, after factoring out the greatest common factor, we have \(x^2 - 4y^2\) left inside the parentheses. Recognize this as a difference of squares:

• \(x^2\) is the square of \(x\)
• \(4y^2\) is the square of \(2y\)

To factor this using the difference of squares:

• Notice \(a = x\) and \(b = 2y\) in \(a^2 - b^2\).
• Apply the formula \(a^2 - b^2 = (a + b)(a - b)\).

This gives us \((x + 2y)(x - 2y)\), nicely factoring the expression into a product of binomials.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are fundamental in algebra, serving as a building block for equations and polynomials.
When working with algebraic expressions such as \(8x^2 - 32y^2\), it is crucial to break them down into simpler components. This usually involves factoring, which makes it easier to understand the structure and relationships within the expression.
In solving the expression in the exercise, we used key algebraic processes:

• Identified a GCF to simplify the expression initially.
• Recognized specific patterns like the difference of squares for further simplification.
• Applied known formulas to factorize efficiently.

Mastering the manipulation of algebraic expressions is an essential skill. It allows you to solve complex problems with ease and prepares you for more advanced math topics.