Problem 58 - Elementary Algebra

Question

Factor out the GCF. $$ 2 x(3 x+4)+(3 x+4) $$

Step-by-Step Solution

Answer

Factor: \((3x+4)(2x+1)\).

Step 1: Identify common factors

Look at the two terms in the expression: the first term is \(2x(3x+4)\), and the second term is \((3x+4)\). Notice that \((3x+4)\) is a common factor in both terms.

Step 2: Factor out the GCF

Since \((3x+4)\) is the greatest common factor (GCF), factor it out from the expression. This means rewriting the expression as \((3x+4)(2x+1)\). To double-check, redistribute \((3x+4)\) to see if you obtain the original expression: \((3x+4)(2x) + (3x+4)(1) = 2x(3x+4) + (3x+4)\), which matches the original expression.

Step 3: Confirm the factored expression

Check that there are no further common factors to pull out from \(2x+1\). With nothing else to factor, the expression \((3x+4)(2x+1)\) is fully factored.

Key Concepts

Understanding Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is a key concept in factoring algebraic expressions. It is the largest factor that divides two or more numbers or terms without leaving a remainder. In algebra, determining the GCF helps simplify expressions and solve equations more easily.
When dealing with algebraic expressions, finding the GCF involves identifying common factors in the terms of the expression.

• First, list the factors of each term in the expression.
• Next, identify the largest factor common to all the terms.
• This common factor is the GCF and can be factored out of the expression.

In our example, \(2x(3x+4)\) and \((3x+4)\) share \((3x+4)\) as a common factor, making it the GCF. Factoring out this common part simplifies the expression.

Exploring Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operational symbols. It's a building block of algebra that helps us model real-world scenarios and solve problems.
For instance, consider the expression \(2x(3x+4) + (3x+4)\). It comprises of terms, coefficients, and variables:

• Terms: The parts of the expression separated by plus or minus signs, such as \(2x(3x+4)\) and \((3x+4)\).
• Coefficients: Numbers that multiply the variables, like \(2\) in \(2x\).
• Variables: Symbols representing unknown values, such as \(x\).

Understanding these components allows us to tackle complex expressions systematically, identify common factors, and manipulate them according to the rules of algebra.

Performing Mathematical Operations in Algebra

Mathematical operations are the heart of algebra. They include addition, subtraction, multiplication, and division. These operations are essential tools for manipulating expressions and equations.
In factoring, multiplication comes into play significantly:

• Identify and factor out the greatest common factor using multiplication and division.
• After removing the GCF, the remaining expression is often further simplified by multiplication and addition.

For example, when we factor out the GCF, \((3x+4)\), from \(2x(3x+4) + (3x+4)\), we essentially use division (to separate \((3x+4)\) from each term), followed by multiplication to check our work: \((3x+4)(2x+1) = 2x(3x+4) + (3x+4)\).
Proficiency in these operations allows for efficient manipulation and simplification of algebraic expressions, making problem-solving smoother.