Problem 53 - College Algebra

Question

In Exercises \(49-56,\) factor using the formula for the sum or difference of two cubes. $$8 x^{3}-1$$

Step-by-Step Solution

Answer

The factored form of \(8 x^{3}-1\) is \((2x - 1)(4x^2 + 2x + 1)\).

Step 1: Identify the Cubes

First, note that \(8 x^{3}\) can be written as \((2x)^{3}\) and \(1\) can be written as \(1^{3}\). So, this is in the form of \(a^3 - b^3\), where \(a = 2x\) and \(b = 1\).

Step 2: Apply the Difference of Cubes Formula

Apply the formula for the difference of cubes which is \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Replacing 'a' with '2x' and 'b' with '1' we get: \((2x - 1)((2x)^2 + (2x)*1 + 1^2)\).

Step 3: Simplify the Result

Simplify the factored terms to arrive at the final solution. The equation becomes \((2x - 1)(4x^2 + 2x + 1)\).

Key Concepts

Difference of Cubes

When you come across a problem involving the difference of cubes, it's important to recognize the pattern quickly. The difference of cubes formula is a handy tool that helps us break down complex expressions into simpler factors. It states that if you have an expression of the form \(a^3 - b^3\), it can be factored into \((a - b)(a^2 + ab + b^2)\).
This is particularly useful when you deal with cubic expressions because it provides a set structure to follow.

• Identify the expression in the form of two cubes.
• Use the formula to factor them.
• Simplify the remaining expression to its simplest form.

In the original exercise, we identified \(8x^3 - 1\) as \((2x)^3 - 1^3\), where \(a = 2x\) and \(b = 1\). Plugging these into the formula makes factoring straightforward and manageable.

Factoring Techniques

Factoring is an essential algebraic process used to simplify expressions and solve equations. Various techniques exist for factoring, each tailored to different types of expressions. Among these, the difference of cubes is just one technique out of many.

When using factoring techniques:

• First, always search for a common factor between terms.
• Check if the expression is a special form, such as a difference of squares or cubes.
• Apply the appropriate special formula.

This systematic approach simplifies even the most complex algebraic expressions. In many exercises, like the one given, recognizing patterns such as the difference of cubes can transform intimidating equations into solvable forms.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators. Understanding how to manipulate these expressions is crucial to mastering algebra. Simplifying and factoring are two primary operations performed on algebraic expressions.

Here’s why mastering algebraic expressions is essential:

• They represent real-world problems mathematically, making them easier to analyze.
• Factoring these expressions can lead to understanding their roots and properties, which is vital for solving equations.
• Breaking down complex expressions into simpler parts can reveal underlying mathematical relationships.

For example, in the exercise \(8x^3 - 1\), identifying it as a difference of cubes allowed us to factor it neatly, demonstrating the power of understanding and manipulating algebraic expressions effectively.