Problem 6 - College Algebra and Calculus: An Applied Approach

Question

Factor out the common factor.\(3 x(x+2)-4(x+2)\)

Step-by-Step Solution

Answer

The expression \(3 x(x+2)-4(x+2)\) can be factored to \((x+2)(3x-4)\).

Step 1: Identify the Common Factor

The first step is to identify the same factor in both terms. Here, the common factor in \(3 x(x+2)\) and \(-4(x+2)\) is \((x+2)\).

Step 2: Factor Out the Common Factor

Next, we'll take out the common factor. To do this, we distribute the terms back without the common factor. The equation \((x+2)\) is present in both terms, hence it becomes: \((x+2)(3x-4)\)

Key Concepts

Common Factor

A common factor is a term or expression that is shared by each term in an algebraic expression. Identifying common factors is one of the fundamental skills in algebra, especially when factoring expressions. Finding a common factor can simplify an expression considerably.
If we take the expression given in the exercise

• \(3x(x+2)-4(x+2)\)

we'll notice that both terms include the expression \((x+2)\). This shared term is what we call the common factor.
To factor out this common factor, you basically "undistribute" it from each of the terms in the expression. Factoring out the common factor simplifies the expression, making it easier to handle in further algebraic manipulations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. The expression given in the exercise

• \(3x(x+2)-4(x+2)\)

is an example of an algebraic expression that combines variables and constants via addition, subtraction, and multiplication.
An algebraic expression does not have an equals sign like algebraic equations. This distinction is crucial as expressions are simplified and equations are solved. Understanding how to work with algebraic expressions by adding, subtracting, or factoring them is essential in solving mathematical problems.
Expressions like the one in this exercise, which involve factoring, help in breaking down complex expressions into simpler parts, revealing patterns and making calculations easier.

Polynomials

Polynomials are a specific type of algebraic expression characterized by terms which are composed of variables raised to whole-number exponents. A polynomial can have one or more terms, and each term consists of a coefficient multiplied by a variable raised to an exponent.
In the original exercise, the expression

• \(3x(x+2)-4(x+2)\)

fits the definition of a polynomial because both of its terms involve the variable \(x\) raised to the first power. When factoring polynomials, like in the given expression, you often look for a common factor across the terms to simplify the expression.
Being familiar with polynomials and strategies for their manipulation, such as factoring, empowers you with the tools to solve more complex algebraic problems easily. The ultimate goal is to achieve simpler expressions like

• \((x+2)(3x-4)\)

which are easier to manage and understand.