Problem 3 - College Algebra

Question

Factor out the greatest common factor. $$ 3 x^{2}+6 x $$

Step-by-Step Solution

Answer

The equation \(3x^{2}+6x\) factored out will be \(3x(x + 2)\).

Step 1: Identify the Greatest Common Factor

To begin with, it's essential to determine the common factor between \( 3x^{2} \) and \( 6x \). Here, the greatest common factor is \(3x\).

Step 2: Factor out the Greatest Common Factor

Next, we factor out the common factor \(3x\) from the equation. Doing this, the equation becomes: \(3x(x + 2)\).

Step 3: Simplify

There is nothing more to simplify here. Therefore, the final factored equation is \(3x(x + 2)\).

Key Concepts

Factoring

Factoring is a foundational concept in algebra that involves expressing a mathematical expression as a product of its factors. It is similar to breaking down a number into smaller numbers that can be multiplied to give the original number. In the context of algebraic expressions, factoring requires identifying components that are common among the terms within the expression.

Consider an expression like \(3x^2 + 6x\). The goal here is to express this polynomial as a product of simpler polynomials or numbers. This is where the concept of the greatest common factor becomes very handy. Factoring effectively simplifies an equation, making it much easier to work with in subsequent algebraic operations.

To successfully factor expressions, you should always first look for the greatest common factor among the terms.

Greatest Common Factor

The greatest common factor (GCF) refers to the highest number and/or variable that can evenly divide every term in the given expression. Starting with finding the GCF is crucial in factoring algebraic expressions. Why? Because it allows you to simplify the problem, making further manipulation of the expression straightforward.

To illustrate, in the problem \(3x^2 + 6x\), the coefficients are 3 and 6 while the variable part is \(x\) common in both terms. Breaking it down:

• For 3 and 6, the GCF is 3 because it is the largest number that divides both.
• For the variable part, \(x\) is present in both \(3x^2\) and \(6x\), so \(x\) itself is considered a part of the GCF.

Hence, the GCF for \(3x^2 + 6x\) is \(3x\). Recognizing and factoring out this GCF is the essential first step in simplifying the expression.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the process of making an expression as concise and straightforward as possible. The aim is to tidy up complex equations into manageable and easily interpretable forms. Once you have factored out the greatest common factor, as seen in the step-by-step solution, you've already simplified the expression considerably.

For the expression \(3x^2 + 6x\), once we factor out \(3x\), we are left with \(3x(x + 2)\). Here, there’s nothing more to simplify, as all parts of the factorization are in their simplest forms.

• \(3x\) is fully simplified, as there are no further common factors to extract.
• \(x + 2\) is in its simplest form since it is a linear expression with no like terms to combine.

Consistently simplifying expressions helps to make them easier to solve or manipulate in broader algebraic equations, reducing the number of steps needed for further operations.