Problem 98 - Elementary Algebra

Question

Make up your own sum or difference of cubes factoring exercise and provide the answer. Explain how you solved it.

Step-by-Step Solution

Answer

The factored form of \( 8x^3 - 27 \) is \( (2x - 3)(4x^2 + 6x + 9) \).

Step 1: Identify the Problem Type

First, determine whether the expression is a sum or difference of cubes. We'll consider the expression \( 8x^3 - 27 \, \). This is a difference of cubes since \( 8x^3 = (2x)^3 \) and \( 27 = 3^3 \).

Step 2: Write in Cube Form

Express both terms in the form of cubes: \( 8x^3 \) becomes \( (2x)^3 \) and \( 27 \) becomes \( 3^3 \). So, rewrite the expression as \( (2x)^3 - 3^3 \).

Step 3: Apply Difference of Cubes Formula

Recall the difference of cubes formula: \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \). Here, \( a = 2x \) and \( b = 3 \).

Step 4: Substitute Values into Formula

Substitute \( a = 2x \) and \( b = 3 \) into the formula: \( (2x - 3)((2x)^2 + (2x)(3) + 3^2) \).

Step 5: Simplify the Expression

Simplify the terms: \((2x)^2 = 4x^2\), \((2x)(3) = 6x\), and \(3^2 = 9\). So, the factored expression becomes \( (2x - 3)(4x^2 + 6x + 9) \).

Key Concepts

Difference of Cubes

Difference of cubes refers to an algebraic expression that takes the form \(a^3 - b^3\). In a difference of cubes, both terms are perfect cubes, making it possible to factor using a special formula. This formula is:

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

Let's break this down with an example expression like \(8x^3 - 27\). Our first step is to recognize both terms as perfect cubes:

• \(8x^3 = (2x)^3\)
• \(27 = 3^3\)

Rewriting in cube form gives us \((2x)^3 - 3^3\). Now, identifying \(a\) as \(2x\) and \(b\) as 3 allows us to plug into the formula, producing \((2x - 3)(4x^2 + 6x + 9)\). By using the special formula, you can simplify complex expressions into more manageable factors.

Sum of Cubes

The sum of cubes is an expression that looks like \(a^3 + b^3\). Unlike the difference of cubes, it adds two cube terms and also has a specific factoring formula:

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

For example, let's consider the expression \(x^3 + 64\). Here, we see:

• \(x^3 = x^3\)
• \(64 = 4^3\)

These are perfect cubes, \(x\) and \(4\). Substitute \(a = x\) and \(b = 4\) into the formula to get \((x + 4)(x^2 - 4x + 16)\). Breaking it down this way helps make sense of the sum of cubes and its conversion into simpler factors.

Factoring Polynomials

Factoring polynomials involves breaking down complex algebraic expressions into simpler products of polynomials. It can simplify expressions and solve equations by finding their roots. Different techniques exist for factoring, including using formulas specifically for cubes.

When dealing with polynomials like cubes, you have to recognize whether it is a sum or difference of cubes. Here’s how to approach it:

• Identify each term as a perfect cube.
• Use relevant formulas:
  - Difference: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
  - Sum: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
• Simplify further as necessary.

Applying these steps to the case of either sum or difference, such as \(8x^3 - 27\) (a difference), helps illustrate how polynomials can be compacted into neater factors. Understanding these concepts and formulas is fundamental to mastering algebraic expressions.