Problem 33 - Algebra 1: Concepts and Skills

Question

Factor the expression. \(c^{3}-8\)

Step-by-Step Solution

Answer

The factorized form of \(c^{3}-8\) is \( (c-2)(c^{2}+2c+4)\).

Step 1: Identify the formula

For this exercise, we need to identify that we can use the difference of cubes formula. The formula is \(a^{3}-b^{3} = (a-b)(a^{2}+ab+b^{2})\). Here, a and b should be chosen such that they form \(c^{3}\) and \(8\) respectively.

Step 2: Identify the values of a and b

In this case, 'a' would be equal to 'c' as \(c^{3}\) is a perfect cube and 'b' would be 2 because \(2^{3} = 8\). Therefore, we can replace 'a' with 'c' and 'b' with 2 in the formula.

Step 3: Apply the formula

We substitute 'a' with 'c' and 'b' with 2 in the formula \(a^{3}-b^{3} = (a-b)(a^{2}+ab+b^{2})\). Thus, we get \(c^{3}-8 = (c-2)(c^{2}+2c+4)\). This is the factorized form of the given expression.

Key Concepts

Difference of Cubes

In algebra, the difference of cubes is a popular concept for simplifying expressions involving cubic terms. It gives one a structured way to factor expressions like \(a^3 - b^3\). The special formula used is:

• \(a^3 - b^3 = (a-b)(a^2+ab+b^2)\)

This formula is applicable when you have two perfect cubes arranged as a subtraction.
For any numbers or variables to fit this formula, each term must be expressible as a cube. In our example \(c^3 - 8\), we recognize:

• \(c^3\) is \((c)^3\)
• \(8\) is \(2^3\)

By identifying \(a\) as \(c\) and \(b\) as 2, you can factor the expression using the formula efficiently.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. In this context, it's about using algebraic structures to solve equations and factor expressions.
When working with the difference of cubes, you are employing fundamental algebraic techniques. Simplifying and factoring are key operations in algebra, and you can accomplish them by careful observation and the application of relevant formulas. In this exercise, algebra involves identifying \(c\) and \(2\) as the terms to fit into the difference of cubes framework.
Using algebraic skills, you can transform the expression \(c^3 - 8\) into \((c-2)(c^2 + 2c + 4)\), which is simpler and often easier to work with in further calculations.

Polynomials

Polynomials are mathematical expressions consisting of variables raised to various powers, combined using addition, subtraction, and multiplication. An expression like \(c^3 - 8\) is a polynomial because it contains a power of three.
Factoring polynomials is essential, particularly when dealing with equations you need to solve or simplify for further operations. By expressing \(c^3 - 8\) as \((c-2)(c^2 + 2c + 4)\), you break it down into simpler linear and quadratic terms.
This not only helps in simplifying the polynomial further, where required, but also in solving algebraic equations where such expressions occur. Recognizing that \(c^3 - 8\) is a polynomial encourages you to use techniques such as the difference of cubes to gain more insights and simplify your work.