Problem 63 - College Algebra Essentials

Question

Factor using the formula for the sum or difference of two cubes $$27 x^{3}-1$$

Step-by-Step Solution

Answer

The factored form of the given expression \(27x^3 - 1\) using the formula for the difference of two cubes is \( (3x - 1)(9x^2 + 3x + 1)\).

Step 1: Identify a and b

First identify \(a\) and \(b\) for which \(a^3 = 27x^3\) and \(b^3 = 1\). Hence in this case, \(a = 3x\) (since \((3x)^3 = 27x^3)\) and \(b = 1\) (since \(1^3 = 1)\).

Step 2: Apply the Difference of Cubes Formula

Now, use the formula for the difference of two cubes, which is, \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Applying the same formula for our \(a\) as 3x and \(b\) as 1, we get \( (3x - 1)((3x)^2 + 3x*1+ 1^2)\).

Step 3: Simplify the expression

Simplify the expression obtained in the previous step to get the final factored form. Thus the expression becomes \( (3x - 1)(9x^2 + 3x + 1)\).

Key Concepts

Difference of Cubes

Let's start by understanding what a difference of cubes is in algebra. The term "difference of cubes" refers to an expression that can be written in the form \(a^3 - b^3\). This type of expression can be factored using a special formula. The formula for factoring the difference of two cubes is:

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

In this formula:

• \(a\) and \(b\) are each terms that are being cubed.
• The expression is split into a linear term \((a - b)\) and a quadratic trinomial \((a^2 + ab + b^2)\).

Using this formula, we can decompose any difference of cubes into a smaller, potentially more manageable set of algebraic expressions.
Once factored, solving or simplifying the expression becomes much simpler. Remembering this formula and recognizing expressions in this form is a valuable skill when working with polynomials.

Sum of Cubes

The sum of cubes is another important algebraic concept closely related to the difference of cubes. Here, the expression takes the form \(a^3 + b^3\). While at first glance this might look similar to the difference of cubes, it uses a different formula for factoring:

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

Understanding this formula is crucial as it helps break down the expression into a product of a binomial \((a + b)\) and a quadratic trinomial \((a^2 - ab + b^2)\).
This decomposition aids in simplifying expressions or finding roots more easily. Though the difference of cubes typically shows up more often, the sum of cubes occurs frequently enough to warrant familiarity with this factorization method. Recognizing and applying these formulas correctly can significantly streamline problem-solving process involving polynomial expressions.

Algebraic Expressions

Algebraic expressions, such as those involving cubes we have discussed, form the foundation of algebra. They are combinations of numbers, variables, and operations that represent some quantity. When dealing with polynomials like cubes, the key is to identify and use the right formulas to simplify the expressions.
Working with algebraic expressions can involve several operations:

• Identifying patterns, such as the difference or sum of cubes.
• Applying appropriate formulas for factorization.
• Simplifying the expressions to make them easier to work with.

For the exercise at hand, we focused on recognizing the pattern as a difference of cubes, which led us to apply the respective formula to factor the expression \(27x^3 - 1\). Understanding these patterns in algebraic expressions and knowing how to work with them prepares you for more complex algebraic challenges.