Problem 64 - College Algebra Essentials

Question

Factor using the formula for the sum or difference of two cubes $$8 x^{3}+125$$

Step-by-Step Solution

Answer

The factorized expression of \(8x^{3} + 125\) using the sum of cubes is \((2x + 5)(4x^{2} - 10x + 25)\).

Step 1: Identify the Cubes

Rewrite the given expression by expressing each term as a cube, using the identity given. So \[ 8x^{3}+125 = (2x)^{3} + 5^{3}\].

Step 2: Apply the formula for the sum of cubes

Apply the sum of cubes formula, where \(a\) is \(2x\) and \(b\) is \(5\). So the expression becomes: \[ (2x + 5)((2x)^{2} - (2x)(5) + 5^{2})\].

Step 3: Simplify the obtained expression

The obtained expression simplifies to \((2x + 5)(4x^{2} - 10x + 25)\].

Key Concepts

Factoring Polynomials

Factoring is a critical skill in algebra that involves finding an expression that, when multiplied together, yields the original polynomial. The problem at hand is an example of factoring a sum of cubes, which is a special case. The sum of cubes refers to an expression of the form \( a^3 + b^3 \). When you encounter a polynomial that can be expressed as a sum of two cubes, it can be factored into a binomial times a trinomial. In our exercise, \( 8x^3 + 125 \) is recognized as \( (2x)^3 + 5^3 \). Here, we applied the factoring technique to break down the polynomial into the product of \( (2x + 5) \) and \( (4x^2 - 10x + 25) \), which are much simpler expressions.

The skills acquired in factoring polynomials are foundational for understanding more complex algebraic operations and provide a basis for solving polynomial equations.

Algebraic Identities

Algebraic identities are pre-established formulas that relate algebraic expressions in a way that they can be easily manipulated and simplified. One such identity is the formula for the sum of cubes: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). It's essential to remember this identity, as it provides a quick way to factor an expression that may initially seem daunting. In our original exercise, this identity is used to factor the expression \( 8x^3 + 125 \). Recognizing the structure of algebraic identities and knowing when to apply them can vastly simplify the process of manipulating algebraic expressions. They are extremely useful in various aspects of algebra, including simplifying expressions, solving equations, and understanding polynomial functions.

Simplifying Expressions

Simplifying expressions is about rewriting them in the most concise and efficient form without changing their value. It often involves recognizing and applying algebraic identities. For instance, the original problem \( 8x^3 + 125 \) requires simplification via factoring. After applying the sum of cubes formula, we obtain \( (2x + 5)(4x^2 - 10x + 25) \). Each step in simplifying this expression makes the problem easier to understand and manage. Simplification isn't just about making expressions shorter; it can also involve rearranging terms to reveal characteristics of the expression that can be useful in further calculations or graphing. Students learning to simplify expressions become better equipped to tackle complex algebraic problems.

College Algebra

College algebra is a course that often serves as a foundation for higher-level mathematics and various applications in science, engineering, and beyond. It extends beyond the algebra learned in high school and involves more complex equations, functions, and systems of equations. The original problem falls under the umbrella of college algebra, which includes advanced factoring techniques such as factoring the sum of cubes. Mastery of these concepts is not only critical for academic success in mathematics but also for developing logical thinking and problem-solving skills that are highly valuable in a wide array of academic disciplines and professional fields. As part of college algebra, understanding how to navigate through polynomial equations and their factoring will often play a role in more complex analytical tasks.