Problem 110 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(125 x^{3}-1\)

Step-by-Step Solution

Answer

\((5x - 1)(25x^2 + 5x + 1)\)

Step 1: Identify the form

The given expression is a difference of cubes because it can be represented as \(a^3 - b^3\), where \(a = (5x)\) and \(b = 1\). Thus, we have \(125x^3 = (5x)^3\) and \(1 = 1^3\).

Step 2: Apply the difference of cubes formula

Use the formula for the difference of cubes: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Plug in the values of \(a = 5x\) and \(b = 1\), so the expression becomes \((5x-1)((5x)^2 + (5x)(1) + 1^2)\).

Step 3: Simplify the terms inside the second factor

Calculate each part of the second factor:1. \((5x)^2 = 25x^2\)2. \((5x)(1) = 5x\)3. \(1^2 = 1\)Combine these to get \(25x^2 + 5x + 1\).

Step 4: Write the final factored form

Based on the calculations, the completely factored form of the expression is \((5x - 1)(25x^2 + 5x + 1)\).

Key Concepts

Difference of Cubes

When you see an expression like the one in the original exercise, the format may seem daunting at first, but recognizing it as a difference of cubes simplifies things considerably. A difference of cubes takes the form \(a^3 - b^3\).
To factor expressions in this form, we use the special algebraic formula:

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

Using this formula allows us to break a complex cubic expression into simpler polynomial factors. In our exercise, the expression \(125x^3 - 1\) initially appears complex but becomes straightforward once identified as \((5x)^3 - 1^3\). By recognizing \(a = 5x\) and \(b = 1\), the expression becomes something much easier to handle.

Polynomial Expressions

Polynomial expressions are combinations of variables and constants using operations such as addition, subtraction, multiplication, and non-negative integer exponents. The original exercise features a third-degree polynomial, otherwise known as a cubic polynomial.
These expressions differ in name (linear, quadratic, cubic, etc.) based on their degree, which is the highest power of the variable.
Here, \(125x^3\) is a term of degree 3 and is a critical element in identifying the expression as a difference of cubes. Polynomials can often be factored or simplified using specific formulas or methods such as sum or difference of cubes, as seen in this exercise.

Algebraic Formulas

Algebraic formulas are essentially shortcuts or pre-established rules expressed in algebraic terms that aid in solving problems efficiently. These formulas can help factor complicated polynomial expressions into simpler parts.
Using the difference of cubes formula in this exercise:

• First, we determine \(a = 5x\) and \(b = 1\), which match the structure \(a^3 - b^3\).
• Then we apply the formula \((a-b)(a^2 + ab + b^2)\) to obtain factors \((5x - 1)\) and \((25x^2 + 5x + 1)\).

Learning and understanding these formulas not only streamlines the problem-solving process but also builds foundational skills for tackling more complex algebraic challenges later on.