Problem 54 - A Graphical Approach to College Algebra

Question

Factor each sum or difference of cubes completely. $$8 m^{3}-27 n^{3}$$

Step-by-Step Solution

Answer

\((2m - 3n)(4m^2 + 6mn + 9n^2)\)

Step 1: Identify the Formula

We recognize that the given expression \(8m^3 - 27n^3\) is a difference of cubes. The formula for difference of cubes is \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\).

Step 2: Find the Values of a and b

First, express each term as a cube. For \(8m^3\), we have \((2m)^3\), and for \(27n^3\), we have \((3n)^3\). Thus, \(a = 2m\) and \(b = 3n\).

Step 3: Apply the Difference of Cubes Formula

Substitute \(a = 2m\) and \(b = 3n\) into the formula: \((2m - 3n)((2m)^2 + (2m)(3n) + (3n)^2)\).

Step 4: Simplify the Expression

Calculate the expression: \((2m)^2 = 4m^2\), \((2m)(3n) = 6mn\), and \((3n)^2 = 9n^2\). Substitute these values into the expression to get \((2m - 3n)(4m^2 + 6mn + 9n^2)\).

Step 5: Write the Final Answer

The completely factored form of \(8m^3 - 27n^3\) is \((2m - 3n)(4m^2 + 6mn + 9n^2)\).

Key Concepts

Factoring

Factoring in algebra is the process of breaking down a complex expression into simpler components that, when multiplied together, give the original expression. Think of factoring like unwrapping a present to reveal the smaller, more tangible parts inside. In the context of the problem we're examining, our task is to factor a difference of cubes.The expressions of the form \(a^3 - b^3\) are known to be a difference of cubes, and they can always be factored using the formula:

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

In our example, \(8m^3 - 27n^3\), both terms are perfect cubes. The key to factoring them lies in correctly identifying the values of \(a\) and \(b\) that satisfy \(a^3 = 8m^3\) and \(b^3 = 27n^3\). Once you identify \(a = 2m\) and \(b = 3n\), you use the difference of cubes formula to obtain a fully factored form of the original expression. By practicing factoring, you develop a deeper understanding of how expressions are constructed and can skillfully manipulate them in various algebraic equations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations that represent a particular value or set of values. An expression like \(8m^3 - 27n^3\) involves variables \(m\) and \(n\), making it an algebraic expression.The study and manipulation of algebraic expressions are foundational skills in algebra. This particular expression is special because it involves cubes, making it a candidate for special factoring techniques, such as the difference of cubes method. In practice, algebraic expressions can range from simple linear expressions to complex polynomial forms.When working with algebraic expressions:

• Always look for opportunities to simplify.
• Recognize any patterns or identities, like difference of cubes, to simplify the factoring process.
• Keep track of variables and coefficients to ensure accurate manipulation without altering the expression's intended value.

Understanding how to interpret and process these expressions is a key step in mastering algebra, unlocking the ability to solve equations and model real-world phenomena through mathematics.

Polynomial Factorization

Polynomial factorization involves expressing a given polynomial as a product of its factors, which are polynomials of lower degree. It's akin to breaking down a big puzzle into smaller pieces that are easier to manage and analyze.For polynomials like \(8m^3 - 27n^3\), recognizing that this is a difference of cubes allows you to apply the specific formula for factoring: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). This formula is specifically designed to handle cubes and is a valuable tool in polynomial factorization. Here’s a quick guide to factorizing difference of cubes:

• Identify the cube terms and write them in the form \(a^3 - b^3\).
• Use the formula directly to find the factorized form.
• Check your work by expanding the factors to ensure they multiply back to the original polynomial.

Mastering polynomial factorization opens up possibilities to solve higher-degree polynomial equations and aids in the understanding of polynomial functions’ properties, such as their roots and behavior at the endpoints.