Problem 91 - College Algebra with Modeling and Visualization

Question

Factor the expression. \(x^{3}-1\)

Step-by-Step Solution

Answer

\((x - 1)(x^2 + x + 1)\)

Step 1: Identify the Type of Expression

The expression given is a difference of cubes since it is in the form of \(a^3 - b^3\). Here, \(a = x\) and \(b = 1\).

Step 2: Use the Difference of Cubes Formula

Recall the formula for factoring a difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).

Step 3: Substitute Values into the Formula

Substitute \(a = x\) and \(b = 1\) into the formula: \(x^3 - 1 = (x - 1)(x^2 + x \cdot 1 + 1^2)\).

Step 4: Simplify the Expression

Simplify the expression: \((x-1)(x^2 + x + 1)\).

Step 5: Final Answer

The factorized form of \(x^3 - 1\) is \((x - 1)(x^2 + x + 1)\).

Key Concepts

Difference of Cubes

The difference of cubes is a special type of algebraic expression that can be factored using a specific formula. The expression takes the form \(a^3 - b^3\), where \(a\) and \(b\) are any algebraic expressions. The "difference" refers to the subtraction sign between the two cubes. What makes this expression easier to handle is that it can be broken down into a product of a binomial and a trinomial using the formula:

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

To apply this formula correctly, you must identify \(a\) and \(b\) in the expression you want to factor. Once identified, plug these values into the formula, which gives you the factored form. For example, in the expression \(x^3 - 1\), \(a\) is \(x\) and \(b\) is \(1\). Using the difference of cubes formula, you get \((x - 1)(x^2 + x + 1)\). The simplicity of this process makes the difference of cubes a powerful tool in solving algebraic expressions.

Algebraic Expressions

Algebraic expressions are a combination of constants, variables, and operators such as addition, subtraction, multiplication, and division. These expressions can take many forms, from simple sums like \(x + y\) to more complex forms such as \(x^3 - 1\).

Understanding algebraic expressions is crucial in algebra as they represent real-world quantities and relationships. They allow us to generalize numbers and operations, enabling us to work with unknown values.

When dealing with algebraic expressions, it is essential to understand how to manipulate them. This includes factoring, which is the process of breaking down expressions into their simpler components. This process often uses patterns like the difference of cubes to streamline calculations and solutions. In our original exercise, the expression \(x^3 - 1\) is an algebraic expression that can be factorized into \((x - 1)(x^2 + x + 1)\) through the difference of cubes method. Such techniques highlight the utility and flexibility of algebraic expressions in problem-solving.

Step-by-Step Problem Solving

Step-by-step problem solving is an effective approach when dealing with complex mathematical problems, including factoring algebraic expressions. Breaking down a problem into smaller, manageable steps helps clarify the process and makes tackling tough math problems less intimidating.

In our example of factoring \(x^3 - 1\), we used a systematic method:

• Step 1: Identify the type of expression - Recognize that the expression is a difference of cubes.
• Step 2: Use the appropriate formula - Apply the difference of cubes formula: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
• Step 3: Substitute the values into the formula - Let \(a = x\) and \(b = 1\) to find the expression \((x - 1)(x^2 + x + 1)\).
• Step 4: Simplify as needed - Ensure the expression is fully simplified.

Following these steps helps ensure accuracy and builds a deeper understanding of the underlying algebraic principles. By mastering the step-by-step approach, you will become more efficient in solving various algebraic problems.