Problem 90 - Beginning and Intermediate Algebra

Question

Factor completely. $$125 m^{3}-27 n^{3}$$

Step-by-Step Solution

Answer

\(125m^3 - 27n^3 = (5m - 3n)(25m^2 + 15mn + 9n^2)\)

Step 1: Identify values of a and b

First, we need to identify the values of \(a\) and \(b\). For our expression, we have \(125m^3\) and \(-27n^3\), so we can rewrite it as \((5m)^3 - (3n)^3\), with \(a = 5m\) and \(b = 3n\).

Step 2: Apply the difference of cubes formula

Now that we have the values of \(a\) and \(b\), we can apply the formula for the difference of cubes: \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\] In our case, \(a = 5m\), and \(b = 3n\). Plugging these values into the formula, we get: \[(5m - 3n)((5m)^2 + (5m)(3n) + (3n)^2)\]

Step 3: Simplify the expression

Now, we need to simplify the expression by calculating the squares and the product: \[ (5m - 3n)(25m^2 + 15mn + 9n^2)\] This is the final factored form, and the solution to the exercise is: \(125m^3 - 27n^3 = (5m - 3n)(25m^2 + 15mn + 9n^2)\)

Key Concepts

Difference of Cubes

To solve expressions involving the difference of cubes, it is critical to understand the difference of cubes formula. This formula is a valuable tool in simplifying expressions that fit the pattern of one cube minus another cube. The formula is:\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]Key points to remember:

• Identify values: Check for values of \(a\) and \(b\) such that each part of the expression can be expressed as a cube.
• Apply the formula: With the values identified, apply the difference of cubes formula to factor the expression.

By following these steps precisely, you can transform complex cubic expressions into simpler linear and quadratic terms. It's a fundamental tool for making algebraic expressions more manageable and can be easily memorized and applied.

Polynomials

Polynomials are expressions consisting of variables and coefficients, united by operations such as addition, subtraction, and multiplication. These expressions are divided into terms, where each term has a variable raised to a non-negative integer power. Polynomials can be classified by the number of terms or by the highest power of its variable:

• Monomials: Contain only one term, such as \(5m\).
• Binomials: Consist of two terms, like \(5m - 3n\). This is exactly what we need using the difference of cubes method.
• Trinomials: Have three terms, for example, \(25m^2 + 15mn + 9n^2\) from our solution.

Understanding polynomials is essential in factoring because you often break down an expression into smaller polynomial components. Identifying the type of polynomial at each step helps in choosing the correct method for simplification and solution.

Simplifying Expressions

Simplifying expressions is the manifestation of breaking down complex algebraic expressions into simpler or more manageable forms. This is typically achieved by canceling out terms, factoring components, or combining like terms. In the context of the exercise, simplifying involves several stages:

• Expression Breakdown: Break down the complex expression into simpler components by recognizing patterns or using known formulas, such as the difference of cubes.
• Applying Formulas: Use appropriate algebraic identities; in this case, the difference of cubes formula helps in splitting the expression into a product of a binomial and a trinomial.
• Calculation: Compute expressions like \((5m)^2\) and \((3n)^2\), or \(5m \cdot 3n\) to simplify the terms further.

Through these steps, complicated expressions become more workable. Simplifying expressions is an important skill in algebra, assisting in problem-solving and preparing for more advanced mathematics.