Problem 60 - Elementary and Intermediate Algebra

Question

Factor difference of cubes. \(8 c^{6}-343 w^{3}\)

Step-by-Step Solution

Answer

The factorization is \((2c^2 - 7w)(4c^4 + 14c^2w + 49w^2)\).

Step 1: Identify the Difference of Cubes Formula

The difference of cubes formula is given by: \[a^3 - b^3 = (a-b)(a^2+ab+b^2)\]. We need to express the given expression in this form.

Step 2: Write Expression in Terms of Cubes

Recognize that \(8c^6\) can be rewritten as \((2c^2)^3\) and \(343w^3\) can be rewritten as \((7w)^3\). Thus, the expression can be written as: \[(2c^2)^3 - (7w)^3\].

Step 3: Apply the Difference of Cubes Formula

Substitute \(a = 2c^2\) and \(b = 7w\) into the difference of cubes formula: \[(2c^2 - 7w)((2c^2)^2 + 2c^2 \cdot 7w + (7w)^2)\].

Step 4: Simplify Each Term

Now, simplify the terms inside the second factor: 1. \((2c^2)^2 = 4c^4\)2. \(2c^2 \cdot 7w = 14c^2w\)3. \((7w)^2 = 49w^2\)Thus, the second factor becomes \(4c^4 + 14c^2w + 49w^2\).

Step 5: Write the Final Factorized Form

Combine the results from the previous steps to write the final factorized form: \[(2c^2 - 7w)(4c^4 + 14c^2w + 49w^2)\].

Key Concepts

Difference of Cubes

In algebra, the difference of cubes is a special case of binomial factoring, particularly when we deal with terms that appear in the form \(a^3 - b^3\). This formula is useful because it allows us to break down complex expressions into simpler factors, making them more manageable to work with.

The key formula to remember here is:

• \(a^3 - b^3 = (a-b)(a^2+ab+b^2)\)

This formula essentially states that any expression that can be expressed as the difference of two perfect cubes can be factored into the product of a binomial \((a-b)\) and a trinomial \((a^2+ab+b^2)\).

Using this formula in practice involves correctly identifying the perfect cubes within an expression. For example, in the expression \(8c^6 - 343w^3\), we rewrite it as \((2c^2)^3 - (7w)^3\). Identifying \(2c^2\) and \(7w\) as \(a\) and \(b\) respectively leads us to easily apply the difference of cubes formula.

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as the product of its factors. These factors are polynomials themselves or even simpler expressions. Understanding how to factor polynomials is crucial because it simplifies problems, making them easier to solve or analyze.

Let's break down the factorization step:

• Recognize patterns: Polynomials like \(a^3 - b^3\) can be tackled using specific formulas like the difference of cubes.
• Apply formulas: Use established algebraic identities to convert polynomials into products of simpler expressions.
• Simplify: Once the product form is achieved, simplify each factor to its simplest form.

This approach allows us to take a daunting polynomial and break it into bite-sized pieces.

In the example \(8c^6 - 343w^3\), the polynomial is expressed as two cubes: \((2c^2)^3\) and \((7w)^3\). Applying the difference of cubes factorization as described, it becomes \((2c^2 - 7w)(4c^4 + 14c^2w + 49w^2)\). This process exemplifies polynomial factorization, as it turns a complicated expression into a more digestible form.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operation symbols that represent mathematical values. They form the building blocks of algebra and can take various forms such as monomials, binomials, trinomials, and polynomials.

Understanding how algebraic expressions work is key to solving equations and factoring:

• Monomials: Expressions with a single term, e.g., \(5x\).
• Binomials: Expressions with two terms, e.g., \(a + b\).
• Trinomials: Expressions with three terms, e.g., \(x^2 + 5x + 6\).
• Polynomials: Expressions with multiple terms, such as \(2x^3 - 3x^2 + x - 5\).

The goal is to manipulate these expressions to simplify or solve them, often by factoring or expanding them.

For the expression \(8c^6 - 343w^3\), understanding that it's made up of polynomial terms allows us to employ strategies like recognizing the structure of cubes to factor them. From rewriting it as \((2c^2)^3 - (7w)^3\), it becomes evident how each part corresponds to the components of a cube difference structure.