Problem 41 - Elementary and Intermediate Algebra

Question

Factor. $$ 2 x^{3}+2 $$

Step-by-Step Solution

Answer

The factored form of \(2x^3 + 2\) is \(2(x+1)(x^2-x+1)\).

Step 1: Identify Common Factors

First, look for any common factors in the expression. The terms in the expression are \(2x^3\) and \(2\). Both terms have a common factor of \(2\).

Step 2: Factor Out the Common Factor

Factor out \(2\) from the expression. This gives:\[2(x^3 + 1)\]

Step 3: Recognize a Sum of Cubes

The expression inside the parentheses, \(x^3 + 1\), can be recognized as a sum of cubes. Recall that the formula for a sum of cubes \(a^3 + b^3\) is \((a+b)(a^2-ab+b^2)\).

Step 4: Apply the Sum of Cubes Formula

Apply the formula to \(x^3 + 1^3\):\[(x+1)(x^2-xoldcentered+1)\]

Step 5: Write the Final Factored Form

Combine the results from the previous steps. The completely factored form of the original expression is:\[2(x+1)(x^2-x+1)\]

Key Concepts

Sum of Cubes

When factoring polynomials, recognizing certain patterns can simplify the process significantly. The sum of cubes is one such pattern. For any expression of the form \(a^3 + b^3\), there is a specific formula we can use to factor it effectively. This formula is:

• \(a^3 + b^3 = (a+b)(a^2-ab+b^2)\)

This is especially helpful because it breaks down a complex cubic expression into a product of a binomial and a trinomial. In the given problem, \(x^3 + 1\) is recognized as a sum of cubes with \(x\) as \(a\) and \(1\) as \(b\). Applying this formula, we factor the expression into \((x+1)(x^2-x+1)\). Using the sum of cubes formula helps us avoid lengthy algebraic manipulations and provides a quick route to the factored form.

Common Factor

A common factor in algebraism is a number or variable that divides all terms in an expression without a remainder. Identifying a common factor is often the first step in simplifying expressions since it can reduce the complexity. In our example, the expression is \(2x^3 + 2\). Both terms of this expression share a common factor of \(2\). So, we first factor out \(2\), resulting in \(2(x^3 + 1)\). By recognizing and factoring out this common factor, the expression becomes much simpler and easier to work with.For any polynomial, always start by looking for a common factor. This step often makes subsequent factoring steps quicker and more straightforward.

Algebraic Expressions

At the heart of algebra are algebraic expressions, which are combinations of constants, variables, and arithmetic operations. In our exercise, we were tasked with factoring an expression, which involved both algebraic and arithmetic skills. The original expression \(2x^3 + 2\) is an algebraic expression consisting of two terms. It combines numbers and a variable raised to a power. Keywords in this domain include:

• **Terms**: separated by + or - signs.
• **Coefficients**: numbers in front of the variables, like the \(2\) in \(2x^3\).
• **Exponents**: indicating the power to which a variable is raised, like the \(^3\) in \(x^3\).

Manipulating algebraic expressions involves operations like factoring, expanding, or simplifying. Understanding the components of these expressions is critical for applying various algebraic techniques effectively.