Question

Factor the sum or difference of cubes.\(27 x^{3}+8\)

Short answer

The result is \( (3x+2)(9x^{2}-6x+4) \).

Step-by-step solution

Identify the Cubes

First, identify the cubes in the given expression. We have \(27x^{3}\) and \(8\). In this case, \(27x^{3}\) is the cube of \(3x\) (since \((3x)^{3}=27x^{3}\)) and \(8\) is the cube of \(2\) (since \(2^{3}=8\)). So we can rewrite the original expression as \((3x)^{3}+(2)^{3}\).

Apply the appropriate formula

Now that we know the original expression is the sum of two cubes, we can apply the sum of cubes formula: \(a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})\). Here, \(a=3x\) and \(b=2\). Substitute these values into the formula to find \( (3x+2)((3x)^{2}-(3x)(2)+(2)^{2})\).

Simplify the results

Working this out, we get \( (3x+2)(9x^{2}-6x+4) \). This is the factored form of the given expression.

Key concepts

sum of cubes

Factoring the "sum of cubes" is about recognizing an algebraic expression of the form \( a^3 + b^3 \). The method for factoring such an expression relies on the identities related to these sums. The sum of cubes can be written as:

• \( a^3 + b^3 = (a+b)(a^2-ab+b^2) \)

These formulas are tools to deconstruct the cubes into factors that are simpler to handle. In this context, identifying the 'a' and 'b' for both cubes is critical. For example, in simplifying \( 27x^3 + 8 \), one would first rewrite this as \( (3x)^3 + 2^3 \). Identifying these terms correctly allows the application of the sum of cubes formula effectively. This approach helps simplify and solve complex equations easily.

difference of cubes

The "difference of cubes" is closely related to the sum, with a similar yet distinct formula. It is used when factoring expressions of the form \( a^3 - b^3 \). The key formula to remember is:

• \( a^3 - b^3 = (a-b)(a^2+ab+b^2) \)

Once you have identified which terms are cubic in your expression, you can apply the formula directly for simplification. Though the original exercise didn't focus on this type, the concept is fundamental in algebra as the mechanics of factoring are similar, promoting a comprehensive understanding of cubic expressions in general.
It is often beneficial to contrast the two forms (sum vs difference) to better grasp the nuances of each.

algebraic expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition and multiplication) that represent a particular quantity. These come in various forms and complexities and are foundational in algebra.

• Cubic expressions are one such type, involving a variable raised to the third power.
• In the context of this exercise, recognizing how these expressions fit into 'sum' or 'difference' forms is a critical skill.

The manipulation of algebraic expressions often involves applying formulas or identities such as the sum or difference of cubes. Understanding these expressions' structures enables easier simplification and solution of equations. Additionally, the ability to factor them correctly is a core concept, serving as a building block for more advanced mathematical studies.