Problem 79 - A Graphical Approach to College Algebra

Question

Factor by any method. $$125 m^{6}-216$$

Step-by-Step Solution

Answer

The expression factors as \((5m^2 - 6)(25m^4 + 30m^2 + 36)\).

Step 1: Recognize the expression as a difference of cubes

The expression given is \(125m^6 - 216\). Notice that \(125m^6\) can be rewritten as \((5m^2)^3\) and \(216\) can be rewritten as \(6^3\). Thus, the expression is a difference of cubes: \((5m^2)^3 - 6^3\).

Step 2: Apply difference of cubes formula

The difference of cubes formula is \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). In this expression, \(a = 5m^2\) and \(b = 6\). Substituting into the formula gives:\[(5m^2)^3 - 6^3 = (5m^2 - 6)((5m^2)^2 + (5m^2)(6) + 6^2)\]

Step 3: Simplify the expression

Now simplify the expression:1. \((5m^2)^2 = 25m^4\)2. \((5m^2)(6) = 30m^2\)3. \(6^2 = 36\)Thus, the factors of the expression are \((5m^2 - 6)(25m^4 + 30m^2 + 36)\). This is the fully factored form of \(125m^6 - 216\).

Key Concepts

Factoring Expressions

Factoring expressions involves breaking down a complex expression into simpler factors that, when multiplied together, give the original expression. In the realm of algebra, factoring is a vital skill that simplifies expressions, making them easier to manage and solve. It's like taking apart a large, complicated puzzle into manageable pieces. By doing so, we can better understand the puzzle's structure and find solutions more efficiently. In our example, we have the expression \(125m^6 - 216\). By recognizing parts of the expression, we were able to factor it using the difference of cubes method. This method broke it down into \((5m^2 - 6)(25m^4 + 30m^2 + 36)\). This not only simplified the expression but also made it easier to handle in further algebraic operations.

Polynomial Factorization

Polynomial factorization is a method used to express a polynomial as a product of its factors. It is often employed to simplify complex polynomials, find zeros, or solve equations. In simpler terms, it's the art of finding what multiplies together to form a polynomial. This process can involve a variety of techniques, one of which is recognizing certain patterns, such as the difference of cubes.

• Simplifies expressions.
• Makes solving equations easier.
• Helps in identifying roots and zeros of polynomials.

In the exercise, identifying the expression \((5m^2)^3 - 6^3\) as a difference of cubes was the key step. We then used a formula specific to this kind of polynomial to break it down into \((5m^2 - 6)(25m^4 + 30m^2 + 36)\). Each step brought us closer to understanding and simplifying the problem's structure.

Algebraic Identities

Algebraic identities are equations that hold true for all values of the variables involved. They play a crucial role in simplifying and solving algebraic expressions and equations. Understanding these identities allows you to apply known patterns in mathematical problems for an easier and quicker solution. For instance, common identities include the difference of squares and the difference of cubes formulas.

• Essential for simplifying expressions.
• Helps in factoring and expanding polynomials.

In our exercise, the algebraic identity used was the difference of cubes formula. This formula, \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\), provided a straightforward way to factor the given expression \(125m^6 - 216\). By recognizing that \((5m^2)^3 - 6^3\) fit this identity, it simplified what could have been a tedious process.