Problem 135 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(27 x^{3}-8\)

Step-by-Step Solution

Answer

The expression factors to \((3x - 2)(9x^2 + 6x + 4)\).

Step 1: Identify the form of the expression

The expression \(27x^3 - 8\) is in the form of a difference of cubes, which can be written as \(a^3 - b^3\), where \(a = 3x\) and \(b = 2\).

Step 2: Recall the formula for difference of cubes

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

Step 3: Substitute the values of a and b

Substitute \(a = 3x\) and \(b = 2\) into the formula. This gives us:\[(3x)^3 - 2^3 = (3x - 2)((3x)^2 + (3x)(2) + 2^2)\]

Step 4: Simplify the terms in the factorization

Simplify each term:- \((3x)^2 = 9x^2\)- \((3x)(2) = 6x\)- \(2^2 = 4\)Put these values back into the expression to get:\((3x - 2)(9x^2 + 6x + 4)\).

Step 5: Write the complete factorization

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

Key Concepts

Factoring

Factoring is the process of breaking down an expression into its simplest components that, when multiplied together, result in the original expression. In the case of the exercise given, we are dealing with the difference of cubes. When you see a structure like \(27x^3 - 8\), it resembles \(a^3 - b^3\), where \(a = 3x\) and \(b = 2\). The goal of factoring is to express this in a product of simpler polynomials.Here's how you break it down:

• Recognize the given expression is a difference of cubes.
• Use the specific formula for the difference of cubes to simplify further.

The formula to remember is \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). You simply plug in the values of \(a\) and \(b\) and simplify the terms. This approach is different from factoring other algebraic expressions, like quadratics, because it involves recognizing specific patterns and applying formulas directly, which streamlines the process.

Polynomials

Polynomials are expressions made up of variables and coefficients, related through operations like addition, subtraction, multiplication, and non-negative integer exponents. Each term in a polynomial consists of a variable raised to a power and multiplied by a coefficient. For example, in the term \(9x^2\), 9 is the coefficient, and \(x^2\) is the variable component.When working with polynomials such as \(27x^3 - 8\), especially those that are binomials, our primary aim is to identify patterns such as the difference of cubes. Recognizing such patterns helps simplify them into more manageable factors. The goal is to express the polynomial as a product of smaller polynomials.Whenever you are dealing with polynomials, especially at higher degrees like cubes, it's crucial to:

• Identify the type of polynomial you're working with.
• Apply the appropriate factoring formula or technique.
• Re-arrange and simplify the expression to reach its simplest form.

Understanding polynomials and how to manipulate them is a fundamental skill in algebra, allowing you to tackle various algebraic expressions with ease.

Algebra

Algebra is the branch of mathematics dealing with symbols and rules for manipulating those symbols. It allows us to solve equations and understand relationships between variables. The process of factoring a polynomial, such as the difference of cubes, is rooted in algebraic techniques.In the context of the given exercise, the principles of algebra help us manipulate the expression \(27x^3 - 8\) into its completely factored form \((3x - 2)(9x^2 + 6x + 4)\). Factoring in algebra involves:

• Identifying structures like the difference of cubes or perfect squares.
• Using algebraic identities to rewrite expressions.
• Simplifying complex expressions into products of more straightforward expressions.

Algebra provides the tools needed to transition from complex expressions to simpler forms efficiently. It is a powerful tool in all fields of mathematics and beyond, forming the foundation for much analysis and problem-solving.