Problem 54 - College Algebra

Question

In Exercises \(49-56,\) 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 \(27x^3 - 1\) is \((3x - 1)(9x^2 + 3x + 1)\).

Step 1: Identify the Cubes

The given expression is \(27x^3 - 1\). Here, \(27x^3\) is the cube of \(3x\), and \(1\) is the cube of \(1\). So, we can rewrite the expression as \((3x)^3 - 1^3\).

Step 2: Apply the Formula for the Difference of Cubes

The formula for the difference of cubes is \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Here, \(a = 3x\) and \(b = 1\). Applying this formula, we get the factored expression \((3x - 1)((3x)^2 + (3x)(1) + 1^2)\).

Step 3: Simplify the Factored Expression

Simplifying each part of the factored expression, we get \((3x - 1)(9x^2 + 3x + 1)\).

Key Concepts

Difference of Cubes

The difference of cubes is a mathematical expression formed by subtracting one cube from another. In algebra, a cube is simply a number raised to the power of three. For example, if you have the expression \(a^3 - b^3\), it represents the difference of cubes where both \(a\) and \(b\) are raised to the power of three.

The formula for factoring the difference of cubes, \(a^3 - b^3\), is given by:

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

Using this formula allows us to break down a complex expression into simpler parts, which makes it easier to handle or further simplify. In our exercise example, the expression \(27x^3 - 1\) can be identified as a difference of cubes where \(27x^3\) is the cube of \(3x\) and \(1\) is the cube of 1.

By recognizing these cube components, we can substitute into the formula and factor the expression correctly. It's important to remember that this formula only works when dealing with cubes, making identifying terms crucial before proceeding with factoring.

Sum of Cubes

Just as the difference of cubes can be factored, so too can the sum of cubes. When you have the expression \(a^3 + b^3\), it represents a sum of cubes. Like in the difference of cubes, both \(a\) and \(b\) are raised to the power of three. This is another important case in algebra that allows us to simplify expressions.

The sum of cubes formula is slightly different from the difference of cubes:

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

This formula uses addition and subtraction in different places to accommodate the opposite operation of addition in \(a^3 + b^3\).

Even though our original exercise did not involve a sum of cubes, understanding this concept gives students a broader perspective on how to factor different cubic expressions. Recognizing whether you are working with a sum or difference of cubes is crucial, as it determines which formula to use for correct factorization. This understanding and the knowledge of these factoring techniques are essential for tackling various algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a particular quantity. They can include constants like numbers, variables that can represent a range of values, and operations such as addition, subtraction, multiplication, and division.

In algebra, we often deal with different forms of expressions ranging from simple linear types to more complicated cubic ones like those seen in difference and sum of cubes problems. The process of factoring is a critical skill that helps unravel these expressions to simpler components which reveals their structure and inter-relations.

By factoring expressions, particularly cubic expressions as shown in the original exercise (\(27x^3 - 1\)), we can transform them into a product of simpler equations. This not only aids in solving equations but also in understanding their behavior and properties.

Mastering the manipulation and transformation of algebraic expressions is foundational in algebra. This empowers students to tackle more advanced mathematical problems with efficiency, demonstrating clear, logical reasoning and problem-solving abilities.