Problem 59 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ 8 x^{3}-125 $$

Step-by-Step Solution

Answer

The completely factored expression is \((2x - 5)(4x^2 + 10x + 25)\).

Step 1: Recognize the Difference of Cubes

The expression given is a difference of cubes in the form \[a^3 - b^3\]where \(a = 2x\) and \(b = 5\) because \( (2x)^3 = 8x^3 \) and \( 5^3 = 125 \). This pattern helps us factor using the difference of cubes formula.

Step 2: Apply the Difference of Cubes Formula

The difference of cubes formula is \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]Substituting \(a = 2x\) and \(b = 5\) into the formula, we get:\[8x^3 - 125 = (2x - 5)((2x)^2 + (2x)(5) + 5^2)\]

Step 3: Simplify Each Part of the Factoring Formula

Further simplify the terms inside the parenthesis:1. \((2x)^2 = 4x^2\)2. \((2x)(5) = 10x\)3. \(5^2 = 25\)Thus, the expression becomes:\[(2x - 5)(4x^2 + 10x + 25)\]

Step 4: Verify the Factored Expression

Double-check each term in the expression \[ (2x - 5)(4x^2 + 10x + 25) \]to ensure it is correct by expanding it back using distribution and confirming it matches the original expression \[8x^3 - 125\]. All terms should accurately give the original polynomial.

Key Concepts

Difference of Cubes

In algebra, the concept of "difference of cubes" is crucial when it comes to factoring certain expressions. A difference of cubes takes the form \(a^3 - b^3\), where \(a\) and \(b\) can be any whole number or algebraic term. This type of expression is recognizable because it involves cubes subtracted from one another.

The difference of cubes can be factored using the formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). This formula is handy because it provides a straightforward way to break down the expression into simpler binomial and trinomial factors.

In the problem at hand, \(8x^3 - 125\), we identify \(2x\) as \(a\) and \(5\) as \(b\). This recognition allows us to apply the difference of cubes formula effectively. Understanding this technique is foundational for working with more complex polynomial expressions.

Algebraic Expressions

Algebraic expressions are a fundamental part of mathematics involving numbers, variables, and operations. They can range from simple, like \(3x + 2\), to more complex forms, such as \(8x^3 - 125\). In our example, the expression represents a polynomial that can be simplified using specific factoring techniques.

An algebraic expression consists of terms, each made of coefficients (numeric factors) and variables (symbolic factors). The expression \(8x^3 - 125\) includes two terms: \(8x^3\) with a variable and \(-125\) as a constant.

Identifying expressions as differences of cubes requires recognizing the structure of each term. In this case, seeing \(8x^3\) as \((2x)^3\) and \(125\) as \(5^3\) allows us to proceed with factoring directly.

Polynomial Functions

Polynomial functions are expressions that comprise sums of multiple terms with non-negative integer exponents. These functions are expressed in the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0\), where the \(a_i\)'s are coefficients and \(n\) is the degree of the polynomial.

The function \(8x^3 - 125\) is a third-degree polynomial due to its highest exponent, 3. It signifies that the function is cubic, which is important for determining the types of solutions and how the function can be factored.

Factoring polynomial functions like these aids in finding their zeros or roots, as solving \(f(x) = 0\) often requires breaking the polynomial into simpler factors. Once a polynomial is factored, these factors simplify the solution process, highlighting the role of factoring in understanding polynomials more thoroughly.

Factoring Techniques

Factoring techniques are essential tools for solving equations and simplifying expressions. The "difference of cubes" is one such technique, among several in algebra, which involves recognizing and simplifying specific polynomial structures.

Factoring often starts by identifying patterns or structures in an expression, like the form \(a^3 - b^3\). Other techniques include finding common factors, using the quadratic formula, or applying special formulas such as sums and differences of cubes or squares.

Successfully factoring \(8x^3 - 125\) into \((2x - 5)(4x^2 + 10x + 25)\) showcases how the correct technique transforms complex expressions into simpler, manageable forms. Mastering these techniques enables solving a wide range of algebraic problems effectively, illustrating the importance of factoring in mathematical practice.