Problem 51 - Algebra for College Students

Question

Use the sum-of-two-cubes or the difference-of-two-cubes pattern to factor each of the following. $$1-27 a^{3}$$

Step-by-Step Solution

Answer

The factored form is \((1 - 3a)(1 + 3a + 9a^2)\).

Step 1: Recognize the Difference of Cubes Pattern

The given expression is a difference: \(1 - 27a^3\). This can be expressed in the form \(x^3 - y^3\), where \(x = 1\) and \(y = 3a\). Recognizing this pattern is crucial for factoring the expression properly using the difference of cubes formula.

Step 2: Recall the Difference of Cubes Formula

The difference of cubes formula states that \(x^3 - y^3 = (x - y)(x^2 + xy + y^2)\). This will be used to factor the expression once we have identified \(x\) and \(y\).

Step 3: Substitute Values into the Formula

Substitute \(x = 1\) and \(y = 3a\) into the difference of cubes formula. This gives us:\[1 - (3a)^3 = (1 - 3a)(1^2 + 1 \cdot 3a + (3a)^2)\]

Step 4: Simplify the Factored Expression

Simplify the terms inside the parentheses after substitution:\[(1 - 3a)(1 + 3a + 9a^2)\] Now, combine the terms in the second parentheses to complete the factoring process.

Key Concepts

Difference of Cubes

The "difference of cubes" is a technique used in algebra to factor expressions like the one given in the exercise. When you encounter a problem such as \(1 - 27a^3\), it's helpful to notice that it fits the pattern of \(x^3 - y^3\). In this case, identify \(x = 1\) (since \(1^3 = 1\)) and \(y = 3a\) (since \((3a)^3 = 27a^3\)). This recognition allows you to use a special factoring formula efficiently.

The difference of cubes can be broken down using the formula:

• \(x^3 - y^3 = (x - y)(x^2 + xy + y^2)\)

This formula simplifies the process by reducing what could be a complex polynomial into two smaller expressions. Understanding this concept provides a powerful tool in simplifying expressions and solving equations. Picking out the "difference of cubes" pattern is often the first step in transforming a challenging algebra problem into something more approachable.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations combined in a meaningful way. In simple terms, they are mathematical phrases that can represent real-world quantities. For example, \(1 - 27a^3\) is an algebraic expression that consists of a constant term "1" and the term "27a^3," where 27 is a coefficient, and "a" is a variable raised to the power of three.

Recognizing types of algebraic expressions, like the one above, is key to navigating more complex algebraic tasks. Important components of these expressions include:

• Coefficients: Numbers that are multiplied by the variables.
• Variables: Symbols like \(a\) that can represent unknown values.
• Exponents: Numbers that tell you how many times to multiply the variable by itself.

Breaking down these parts helps students manage their algebraic expressions more effectively. It's also crucial for identifying specific patterns, like the difference or sum of cubes, and then employing appropriate strategies or formulas to simplify or factor these expressions.

Polynomial Identities

Polynomial identities are like algebraic shortcuts. They offer pre-set formulas that help you factor or simplify polynomials efficiently. Knowing these identities is extremely beneficial, particularly in algebra, since they save time and effort, allowing you to focus on more critical problem-solving steps.

The difference of cubes identity is one such shortcut used in the original exercise. The full identity is:

• \(x^3 - y^3 = (x - y)(x^2 + xy + y^2)\)

This provides a direct way to split a cubic expression into a product of a binomial and a quadratic trinomial. Memorizing and understanding these identities empowers students to handle complex expressions systematically.Moreover, polynomial identities are not just tools for simplifying equations; they also form the foundation for more advanced studies in mathematics. Mastery over polynomial identities enhances your ability to engage with algebraic manipulation confidently and opens doors to more advanced mathematical concepts.