Problem 36 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The factored form of the expression is \((3a - b)(9a^2 + 3ab + b^2)\).

Step 1: Identify the type of expression

The given expression is a difference of cubes, which can be factored using the formula. A difference of cubes takes the form \( a^3 - b^3 \) and can be factored as \((a - b)(a^2 + ab + b^2)\).

Step 2: Rewrite the terms as cubes

Recognize that \( 27a^3 \) and \( b^3 \) are perfect cubes. Specifically, we can write \( 27a^3 = (3a)^3 \) and \( b^3 = b^3 \).

Step 3: Apply the difference of cubes formula

Using the identified cubes \((3a)^3\) and \(b^3\), substitute into the factorization formula: \( (a^3 - b^3) = (a - b)(a^2 + ab + b^2) \). Substitute \( a = 3a \) and \( b = b \) into the formula to get \( (3a - b)((3a)^2 + 3a \cdot b + b^2) \).

Step 4: Simplify the factored expression

Simplify the expression obtained from the formula. The expression becomes \( (3a - b)(9a^2 + 3ab + b^2) \) after calculating \((3a)^2 = 9a^2\), \(3a \cdot b\), and \(b^2\).

Step 5: Final expression

Write down the fully factored expression as \( (3a - b)(9a^2 + 3ab + b^2) \).

Key Concepts

Understanding the Difference of Cubes

The difference of cubes formula is a handy tool for factoring expressions that have the form of one cube subtracted from another. Such expressions look like this: \(a^3 - b^3\). This pattern is special because it can always be expressed in a factored form using the specific formula:

• The formula for the difference of cubes is \((a - b)(a^2 + ab + b^2)\).

Unlike regular quadratic expressions or other polynomial forms, the difference of cubes is unique. This formula helps break down seemingly complex cubic expressions into two simpler expressions. This makes them easier to further manipulate or solve.
By identifying that an expression is indeed a difference of cubes, you set the foundation for efficiently factoring it. Applying this formula allows for the correct rearrangement that factors the original expression into two manageable parts: a linear factor \((a - b)\) and a quadratic factor \(a^2 + ab + b^2\).

Factoring Formula for Difference of Cubes

The factoring formula for the difference of cubes is not only a method, but also a shortcut in algebra. You're essentially given a map that shows how to transform a complex expression into simpler terms.

• The formula states: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).

It directly provides a structure to segregate the polynomial into its simplified parts.
To use this formula, you first need to identify the individual cubes within the original expression. Once that’s accomplished, it’s just about plugging in these values into the formula. This is done by substituting the `\(a\)` and `\(b\)` in the equation with the expressions identified as the cubes.
This formula is invaluable when dealing with polynomials that seem difficult to approach. It relieves the tedium of complex factorization by offering a straightforward path.

Identifying Perfect Cubes in Polynomials

A crucial step in using the difference of cubes formula is recognizing perfect cubes. A perfect cube is any number or variable raised to the power of three.

• The number 27 is a perfect cube because it equals \((3)^3\).
• Similarly, any expression like \(a^3\) is a perfect cube because its cube root is a whole number, \(a\).

To determine if a term is a perfect cube, look for expressions or numbers that can be expressed as some base raised to the third power.
For the example expression \(27a^3 - b^3\), observe how \(27a^3\) can be rewritten as \((3a)^3\), and \(b^3\) remains as such. This recognition is the key first step in using the difference of cubes formula effectively.
Identifying these components correctly ensures that you apply the factoring formula accurately, further simplifying the entire expression cleanly into its factored form.