Problem 41 - College Algebra with Corequisite Support

Question

For the following exercises, factor the polynomials. \(64 x^{3}-125\)

Step-by-Step Solution

Answer

The polynomial factors to \((4x-5)(16x^2 + 20x + 25)\).

Step 1: Recognize the Type of Polynomial

The given polynomial is of the form \(a^3 - b^3\), where \(a = (4x)\) and \(b = 5\). This is a difference of cubes.

Step 2: Apply the Difference of Cubes Formula

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

Step 3: Substitute the Values into the Formula

Substitute \(a = 4x\) and \(b = 5\) into the difference of cubes formula: \((4x-5)((4x)^2 + (4x)(5) + 5^2)\).

Step 4: Simplify the Terms

Simplify each part of the formula:1. \((4x-5)\).2. \((4x)^2 = 16x^2\).3. \((4x)(5) = 20x\).4. \(5^2 = 25\).

Step 5: Write the Final Factored Form

Combine the simplified terms to express the polynomial in its factored form: \((4x-5)(16x^2 + 20x + 25)\).

Key Concepts

Difference of Cubes

The concept of the difference of cubes is a powerful tool in polynomial factorization. It allows us to factor polynomials that appear in the form \(a^3 - b^3\). Here, the polynomial is written as the difference of two perfect cubes. To factor these types of expressions, we use the difference of cubes formula:

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

This formula is straightforward yet essential in simplifying and solving polynomial equations. In our specific exercise, \(64x^3 - 125\), you would identify \(a = 4x\) and \(b = 5\). Recognizing these values as perfect cubes (since \(64x^3 = (4x)^3\) and \(125 = 5^3\)) is the crucial first step.
Understanding this concept makes it easier to apply the formula correctly and factor the expression into two binomials: one linear and one quadratic. This reduces the initial cubic expression into simpler parts, aiding in both solving equations and simplifying complex algebraic expressions.

Factoring Techniques

Factoring techniques are diverse strategies used to rewrite algebraic expressions into a product of simpler expressions. Among these techniques, the difference of cubes is particularly valuable, but several others exist for various polynomial forms. Each technique relies on identifying specific patterns or structures within the polynomial.

• For a difference of cubes, the approach is formulaic, requiring the expression to fit \(a^3 - b^3\).
• For sum of cubes, a different pattern, \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\), applies.

The goal of factoring is always to simplify a polynomial into its constituent factors, making it easier to evaluate or solve.
Using the correct technique depends largely on recognizing the structure of the algebraic expression. In our specific example, recognizing that \(64x^3 - 125\) is a difference of cubes directly guides us to apply the relevant formula, thus facilitating a straightforward solution.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent mathematical relationships. Polynomials, a type of algebraic expression, feature sums of terms with variables raised to whole number exponents. They are a fundamental part of algebra.

• Understanding polynomials involves recognizing their degree, which determines their complexity and the techniques applicable in their manipulation.
• Cubic polynomials involve terms up to \(x^3\), such as \(64x^3 - 125\), and require more complex factorization strategies than linear or quadratic polynomials.

Simplifying algebraic expressions often includes factoring, distributing terms, and simplifying fractions.
In our exercise example, the expression \(64x^3 - 125\) represents a difference of cubes, which isn't immediately recognizable as something simple. Factoring transforms it into a more usable form, providing a clearer understanding and a pathway to solutions, particularly in equations or functions. Recognizing patterns like these is pivotal in simplifying and solving polynomials effectively.