Problem 47 - Elementary and Intermediate Algebra

Question

Factor. $$ 64 m^{3} x-8 n^{3} x $$

Step-by-Step Solution

Answer

The expression factors to \(8x(2m-n)(4m^2 + 2mn + n^2)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, examine the given expression: \(64m^3x - 8n^3x\). Look for common factors in both terms. The GCF here is \(8x\), which is the largest factor that divides both \(64m^3x\) and \(8n^3x\).

Step 2: Factor out the GCF

After identifying the GCF, \(8x\), factor it out from each term in the expression. This gives: \[8x(8m^3 - n^3)\].

Step 3: Recognize and Apply Difference of Cubes

Notice that \(8m^3 - n^3\) has the form of a difference of cubes. Recall the formula for difference of cubes: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Here, \(a = (2m)\) and \(b = n\).

Step 4: Factor the Difference of Cubes

Substitute \(a = 2m\) and \(b = n\) into the difference of cubes formula: \[(2m-n)((2m)^2 + (2m)(n) + n^2) = (2m-n)(4m^2 + 2mn + n^2)\].

Step 5: Combine the Results

Substituting back into the expression from Step 2, we have the fully factored form: \[8x(2m-n)(4m^2 + 2mn + n^2)\].

Key Concepts

Greatest Common Factor (GCF)

The greatest common factor, often abbreviated as GCF, is a fundamental concept when it comes to factoring polynomials. It refers to the largest factor that is common to each term in an algebraic expression.
For example, consider the expression \(64m^3x - 8n^3x\). The first step in factoring is identifying the GCF of the terms:\(64m^3x\) and \(8n^3x\). Both terms share \(8x\) as a factor since\(8\) is the greatest factor that divides both \(64\) and \(8\), and \(x\) is a factor common to both terms. By factoring \(8x\) out of each term, the expression simplifies greatly, making further factoring steps easier to visualize and complete.

Difference of Cubes

The difference of cubes is a special algebraic identity that helps in factoring expressions of the form \(a^3 - b^3\). This identity can be expressed as:

• \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)

In the given problem \(8m^3 - n^3\), we can see it fits the difference of cubes format where \(a = 2m\) and \(b = n\). Recognizing this pattern is vital because it allows us to further decompose and factor more complex polynomial expressions into simpler binomials and trinomials. Applying the difference of cubes formula, the expression \(8m^3 - n^3\) factors into \((2m - n)(4m^2 + 2mn + n^2)\).
This step is crucial as it breaks down the expression into components that are easier to work with in subsequent algebraic processes.

Algebraic Expressions

Algebraic expressions are composed of variables and constants that are connected by operations such as addition, subtraction, multiplication, and division. Understanding how to manipulate these expressions, especially when they become complex, is a key skill in algebra.
The original problem \(64m^3x - 8n^3x\) is an example of an algebraic expression that appears complicated at first. However, by applying methods such as identifying the GCF and recognizing the difference of cubes, this expression can be simplified into a combination of factored terms. Each of these components can then potentially be used in further calculations or problem-solving scenarios.

Mastering the ability to deconstruct and factor algebraic expressions unlocks a clearer path to solving equations and understanding the relationships between variables involved.