Problem 45 - Elementary and Intermediate Algebra

Question

Factor. $$ x^{4}-216 x $$

Step-by-Step Solution

Answer

Factor is \(x(x - 6)(x^2 + 6x + 36)\).

Step 1: Identify Common Factor

The expression given is \(x^{4} - 216x\). Both terms have a common factor of \(x\), so we factor out \(x\) from the expression: \[x(x^3 - 216)\] This means both \(x^4\) and \(-216x\) are divisible by \(x\).

Step 2: Identify Special Forms

The expression inside the parentheses is \(x^3 - 216\). Recognize that \(216\) is a perfect cube, as it is equal to \(6^3\). This means you can express \(216\) as \((6)^3\), making \(x^3 - 6^3\) a difference of cubes.

Step 3: Apply Difference of Cubes Formula

The difference of cubes formula is \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Here, \(a = x\) and \(b = 6\), so apply the formula: \[x^3 - 6^3 = (x - 6)(x^2 + 6x + 36)\] Replace \(x^3 - 216\) with the expression from the formula.

Step 4: Factor Completely

Now, substitute back into our factored expression from Step 1: \[x(x^3 - 216) = x(x - 6)(x^2 + 6x + 36)\] This is the completely factored form, as \(x^2 + 6x + 36\) does not factor further with real coefficients.

Key Concepts

Difference of Cubes

The idea of the "difference of cubes" is a fundamental concept in polynomial factoring. Specifically, it refers to expressions in the form \(a^3 - b^3\). This structure means that both terms are perfect cubes. A "difference of cubes" can be factored easily using a specific formula.
The formula for the difference of cubes is:

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

This formula is powerful because it turns a complex cubic expression into a product of a linear term and a quadratic term. To apply it correctly, first identify the cube roots of the terms, \(a\) and \(b\). For example, in the problem \(x^3 - 6^3\), \(a\) is \(x\) and \(b\) is \(6\). Then simply plug these values into the formula to factor the expression into the desired format. Recognizing differences of cubes helps simplify seemingly intricate expressions through this neat algebraic technique.

Common Factor

Identifying a "common factor" is one of the first steps in simplifying polynomial expressions. It involves finding a factor that is common to all terms in a given expression. In the context of the given expression, \(x^4 - 216x\), both terms include at least one factor of \(x\). This implies that \(x\) is a common factor.
To factor out the common factor, simply divide each term by \(x\) and factor it out of the expression:

• Originally: \(x^4 - 216x\)
• Factor out \(x\): \(x(x^3 - 216)\)

Once the common factor is extracted, the remaining expression becomes simpler to manage and may reveal further factorizable parts, like in our example, showing the difference of cubes. Detecting and factoring out common factors is a key skill to master because it simplifies equations and sets the stage for deeper factorization.

Perfect Cube

The term "perfect cube" refers to a number or expression that can be written as the cube of an integer or another expression. Recognizing perfect cubes is essential in identifying special factoring scenarios, such as the "difference of cubes."
A number like \(216\), which equals \(6^3\), is a perfect cube. Being able to express numbers in this form allows for utilizing specific factoring techniques and formulas, such as when rewriting \(x^3 - 216\) as \(x^3 - 6^3\). This transformation is critical because it lets us then apply the difference of cubes formula effectively.

• For example, identify: \(216 = 6^3\)
• Example expression: \(x^3 - 216 = x^3 - 6^3\)

By viewing numbers or terms as cubes, it opens up simpler ways to process and factor them, making algebraic problems much more approachable. Understanding and recognizing perfect cubes is an essential part of mastering polynomial manipulations.