Problem 31 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ 12 x^{3}+18 x $$

Step-by-Step Solution

Answer

The expression completely factored is \( 6x(2x^2 + 3) \).

Step 1: Identify the Greatest Common Factor (GCF)

First, examine each term in the expression \( 12x^3 + 18x \) to identify the greatest common factor. Both terms contain the factor \( 6x \). Thus, the GCF of the expression is \( 6x \).

Step 2: Factor Out the GCF

Write the expression as the product of the GCF and the remaining terms. Divide each term by \( 6x \) to obtain:\[ 12x^3 + 18x = 6x(2x^2 + 3) \].This is the expression factored by the GCF.

Step 3: Check for Further Factoring

Inspect the expression \( 2x^2 + 3 \) within the parentheses to determine if it can be factored further. Since there are no common factors and it's a simple quadratic expression, it cannot be factored further.

Key Concepts

Understanding the Greatest Common Factor (GCF)

The Greatest Common Factor, often abbreviated as GCF, is the largest factor that two or more numbers or terms share. Finding the GCF is a critical first step in simplifying expressions and is especially helpful when factoring polynomial expressions. Let's break down how to identify and utilize the GCF in the context of polynomial expressions.

To find the GCF:

• List the factors of each coefficient in the terms.
• Identify the smallest power of each shared factor between them.
• The GCF is the product of these common factors.

In the example expression, \(12x^3 + 18x\), we start by examining the numeric coefficients \(12\) and \(18\). The common factors of these numbers are \(1, 2, 3, \text{and} 6\), making \(6\) the greatest among them. Additionally, both terms have at least one factor of \(x\), hence combining to identify the GCF as \(6x\). By understanding and applying the concept of the GCF, one can simplify expressions more efficiently.

Factoring by Grouping

Factoring by grouping is a technique used primarily when dealing with polynomial expressions that have four or more terms. It involves rearranging and grouping terms so that they can be factored separately, and then refactoring the expression as a whole. Even though our original example \(12x^3 + 18x\) doesn't necessitate factoring by grouping, understanding this technique is invaluable for more complex expressions.

Here's a simple roadmap to factor by grouping:

• First, split the expression into groups that can be factored separately. Each group must have a common factor.
• Factor out the common factor from each group.
• If successful, the grouped terms will share a common binomial factor.
• Factor this common binomial out to produce a simplified expression.

Factoring by grouping is a creative problem-solving approach. It emphasizes visualizing an expression in two or more parts, and it is particularly effective for quickly finding the simplest form of more complex polynomials.

Decoding Polynomial Expressions

Polynomial expressions consist of variables raised to whole number powers and coefficients – they can appear intimidating at first glance, but they follow straightforward rules. Understanding, identifying, and manipulating polynomial expressions are key skills in algebra.

Polynomials typically take the general form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(a_n, a_{n-1}, \ldots\), and \(a_0\) are constants, also known as coefficients. The power of any term determines its degree. The term with the highest degree is the leading term, which impacts the polynomial’s behavior most significantly as the variable \(x\) changes.

In our expression \(12x^3 + 18x\),

• the highest degree term is \(12x^3\), making it a third-degree polynomial.
• Both terms share similar characteristics, such as containing the variable \(x\).

Approaching polynomials with these structural insights helps in identifying the best factoring strategy. Whether by finding the GCF or arranging terms for grouping, recognizing these key features can unlock faster problem-solving methods in algebra.