Problem 29 - Elementary and Intermediate Algebra

Question

Factor. $$ a^{3}-27 $$

Step-by-Step Solution

Answer

The factorization is \( (a - 3)(a^2 + 3a + 9) \).

Step 1: Recognize the Difference of Cubes

The expression \( a^3 - 27 \) is a difference of cubes. It can be expressed in the form \( a^3 - b^3 \), where \( a = a \) and \( b = 3 \), since \( 27 = 3^3 \).

Step 2: Use the Difference of Cubes Formula

The formula for factoring the difference of cubes is given by: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \] In this exercise, substitute \( a \) with \( a \) and \( b \) with \( 3 \).

Step 3: Substitute the Values into the Formula

Substituting \( a = a \) and \( b = 3 \) into the formula yields: \[ a^3 - 27 = (a - 3)(a^2 + 3a + 9) \]

Step 4: Write the Factorized Expression

The factorized form of \( a^3 - 27 \) is: \( (a - 3)(a^2 + 3a + 9) \)

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a complex expression into simpler factors that, when multiplied together, give the original polynomial. It’s much like solving a puzzle, where the pieces must fit together just right to recreate the entire picture.
In the case of polynomial expressions in algebra, such as sums and differences of cubes, there are special formulas that help in this process. The formula for the difference of cubes is especially useful because it allows us to factor expressions of the form \(a^3 - b^3\) swiftly. This formula is:

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

Recognizing when a polynomial fits this form is crucial. Once recognized, each part of the formula plays its role:

• The \((a - b)\) part simplifies the expression by splitting the components.
• The \(a^2 + ab + b^2\) part represents a trinomial where the respective terms are combinations of \(a\) and \(b\).

This method helps discuss and solve polynomial expressions more efficiently, unveiling patterns that might not be readily apparent in their original form.

Algebraic Expressions

Algebraic expressions are a fundamental part of mathematics, allowing us to generalize mathematical ideas. These expressions are composed of variables and constants, combined using operations such as addition, subtraction, multiplication, and division.
In algebra, understanding and manipulating these expressions is key to solving equations and other mathematical problems. For example, in the exercise to factor \(a^3 - 27\), the expression includes a variable \(a\) and a constant \(27\). Breaking this down into recognizable parts is the first step in understanding how to factor it.
Every algebraic expression can be simplified or transformed based on established mathematical rules. This includes using identities like the formula for the difference of cubes, which is \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). Recognizing and applying these rules allows students to approach problems systematically and improve their problem-solving skills.

Cubic Equations

Cubic equations are polynomial equations of degree three, the standard form being \(ax^3 + bx^2 + cx + d = 0\). These types of equations play a significant role in algebra and their roots determine important properties of algebraic curves.
For the given problem, we focus on a special form of cubic equation known as the difference of cubes. Here, we specifically deal with the scenario where two perfect cubes are subtracted from one another, such as \(a^3 - 27\). Solutions to these problems are often found using specific formulas that simplify the process significantly.
The difference of cubes formula, \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\), is incredibly useful. By treating \(a^3 - 27\) as a cubic equation, it falls into a familiar category of equations that can be solved with precision. Understanding cubic equations and how to manipulate them is a valuable skill, as these often appear in more advanced mathematical problems. Furthermore, recognizing patterns among polynomials, and specifically cubic forms, can lead to greater insights into solving similar algebraic challenges.