Problem 79 - Elementary Algebra

Question

Factor completely. $$ 27 a 3-8 b 3 $$

Step-by-Step Solution

Answer

The expression \( 27a^3 - 8b^3 \) factors to \((3a - 2b)(9a^2 + 6ab + 4b^2)\).

Step 1: Identify the Type of Expression

The given expression is \( 27a^3 - 8b^3 \). This expression resembles the difference of cubes form \( a^3 - b^3 \), where \( a = 3a \) and \( b = 2b \).

Step 2: Apply the Difference of Cubes Formula

The formula for factoring the difference of cubes is \( a^3 - b^3 = (a-b)(a^2+ab+b^2) \). Here, identify our values: \( a = 3a \) and \( b = 2b \).

Step 3: Substitute into the Formula

Substitute \( a = 3a \) and \( b = 2b \) into the formula: \((3a - 2b)((3a)^2 + (3a)(2b) + (2b)^2)\). This becomes \((3a - 2b)(9a^2 + 6ab + 4b^2)\).

Step 4: Verify the Factored Form

Expand the expression \((3a - 2b)(9a^2 + 6ab + 4b^2)\) to ensure it simplifies back to the original expression \(27a^3 - 8b^3\). This confirms our factorization is correct.

Key Concepts

Difference of Cubes

The difference of cubes is a specific form of polynomial expression where two cubes are subtracted from each other. It looks like this: \( a^3 - b^3 \). Recognizing a difference of cubes is crucial because it leads us to a simple factorization formula. In our original problem, \( 27a^3 - 8b^3 \) was identified as a difference of cubes.
The difference of cubes formula is:

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

To use this formula correctly, identify the cube roots of each term. In our problem:

• \( 27a^3 = (3a)^3 \) where \( 3a \) is the cube root
• \( 8b^3 = (2b)^3 \) where \( 2b \) is the cube root

Applying the formula transforms a complex cubic expression into a simpler product of a binomial and a trinomial.

Factoring Techniques

Factoring techniques involve breaking down expressions into products of simpler expressions. One of the key methods for polynomials is recognizing forms such as the difference of cubes.
When you identify a polynomial expression as a difference of cubes, you can apply the corresponding formula directly. This allows you to change the structure of a complicated polynomial into products that are often easier to manage or further simplify.
In our exercise, by observing \( 27a^3 - 8b^3 \) as \( (3a)^3 - (2b)^3 \), the difference of cubes formula was applied:

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

This turns into:

• \( (3a - 2b)(9a^2 + 6ab + 4b^2) \)

Key to these techniques is not just memorization but understanding why and when to use them, offering you a powerful toolset for handling various polynomial expressions.

Polynomial Expressions

Polynomial expressions include terms consisting of variables raised to various powers and multiplied by coefficients. In algebra, these expressions can range from simple to quite complex. Being able to simplify such expressions through factoring is a valuable skill.
In terms of understanding polynomial expressions, recognizing special forms such as cubes and their differences or sums can be helpful. A polynomial like \( 27a^3 - 8b^3 \) doesn't look simple at first glance. However, by dissecting it using algebraic identities or formulas, like the difference of cubes, it becomes manageable: you reduce complexity and open the door to easier calculations or problem solving.
This kind of factorization is particularly important not only for manual calculations but also in higher mathematics, where simplifying polynomials is required for integrals, derivatives, and solutions to polynomial equations. Embrace these opportunities to explore different techniques and expressions, as they expand your mathematical toolkit.