Problem 50 - A Graphical Approach to College Algebra

Question

Which of the following is the correct complete factorization of \(x^{3}+8 ?\) A. \((x+2)^{3} \quad\) B. \((x+2)\left(x^{2}+2 x+4\right) \quad\) C. \((x+2)\left(x^{2}-2 x+4\right) \quad\) D. \((x+2)\left(x^{2}-4 x+4\right)\)

Step-by-Step Solution

Answer

The correct complete factorization is option C: \((x+2)(x^2 - 2x + 4)\).

Step 1: Recognize the Expression Type

The expression given is \(x^3 + 8\), which is a sum of cubes. The sum of cubes formula is \(a^3 + b^3 = (a+b)(a^2-ab+b^2)\).

Step 2: Identify Variables in the Formula

Rewrite \(x^3 + 8\) as \((x)^3 + (2)^3\). Here, \(a = x\) and \(b = 2\).

Step 3: Apply the Sum of Cubes Formula

Substituting into the formula: \(a^3 + b^3 = (a+b)(a^2-ab+b^2)\), we have \((x + 2)((x)^2 - x \,\cdot \, 2 + 2^2)\).

Step 4: Simplify the Factors

The factor \((x)^2 - x \,\cdot \, 2 + 2^2\) simplifies to \(x^2 - 2x + 4\). So the factorization is \((x + 2)(x^2 - 2x + 4)\).

Step 5: Match with Answer Choices

Compare our result \((x + 2)(x^2 - 2x + 4)\) to the provided options. The correct choice is C: \((x+2)(x^2 - 2x + 4)\).

Key Concepts

Sum of Cubes

When dealing with expressions like \(x^3 + 8\), recognizing it as a 'Sum of Cubes' is crucial. This type of expression follows a specific pattern that makes it easier to factorise. The sum of cubes formula is written as: \[ a^3 + b^3 = (a+b)(a^2-ab+b^2) \] where \(a\) and \(b\) represent different terms in the expression. In our exercise, you would reimagine \(x^3 + 8\) as \(x^3 + 2^3\). Here, you can see that \(a\) is \(x\) and \(b\) is 2. This sets us up perfectly to apply the formula efficiently. This pattern is essential because it simplifies seemingly complex expressions and makes it straightforward to identify factor components. Take note; sum of cubes is characteristically different from the difference of cubes, due to the sign differences within the expression.

Polynomials

Polynomials are algebraic expressions that include terms consisting of variables and coefficients. These are composed of operations such as addition, subtraction, multiplication, and nonnegative integer exponents. For example, in the exercise problem \(x^3 + 8\), you have a polynomial composed of two terms: \(x^3\) and 8.

• Each term in a polynomial can be considered as a power of the variable(s), such as \(x^3\).
• The degree of a polynomial is the highest power of the variable; here it is 3 due to \(x^3\).

Understanding polynomials is fundamental in algebra as they form the basis for various equations and functions. Deriving and manipulating polynomials is a skill that unveils solutions to complex algebraic expressions, like recognizing and deploying the sum of cubes strategy in our task. Being able to work with polynomials allows for tackling not just basic but also higher-degree equations.

Algebraic Expressions

Algebraic expressions bring together numbers, variables, and arithmetic operations. These play a significant role in forming mathematical equations and functions. For example, the expression \(x^3 + 8\) combines a variable, \(x\), raised to a power and a constant term, 8.

• Variables, like \(x\), are symbols that represent unknown numbers or values which algebra seeks to define or manipulate.
• Constants, such as the number 8, are fixed values that do not change within the expression.
• Arithmetic operations performed include addition, subtraction, multiplication, and exponentiation.

Expressions can be transformed or simplified using algebraic formulas and principles, such as the sum of cubes formula. This transformation helps to solve equations or convert complex expressions into simpler forms, which are key in the field of algebra. Fully grasping the nature of algebraic expressions helps in understanding their behavior and ultimately mastering their manipulation.