Problem 8 - College Algebra

Question

In Exercises \(1-10\), factor out the greatest common factor. $$x(2 x+1)+4(2 x+1)$$

Step-by-Step Solution

Answer

The greatest common factor of the expression is factored out to obtain: \((2x+1) \times (x + 4)\).

Step 1 Identify the greatest common factor

Upon examining the given expression \(x\(2 x+1\) + 4 \times (2 x+1)\) it can be observed that \(2x + 1\) is a common factor in both terms.

Step 2 Factor out the greatest common factor

The common factor can be factored out to simplify the expression. We do this by condensing the two terms into one, writing the common factor once, and accounting for the remaining components in parentheses. The expression becomes: \((2x+1) \times (x + 4)\).

Key Concepts

Understanding the Greatest Common Factor

When dealing with algebraic expressions, the greatest common factor (GCF) is the largest factor that can divide each term in an expression without leaving a remainder. Think of it as the expression's best friend, the one that fits evenly into each piece of the puzzle. In the given expression, \(x(2x+1) + 4(2x+1)\), both terms include \(2x + 1\). This expression (\(2x + 1\)) is the common thread, or factor, that both terms share.

By identifying \(2x + 1\) as the GCF, we can simplify our tasks. Factoring means taking out this common piece and rewriting the expression in a simplified way. This is helpful because once the GCF is identified, the expression becomes easier to work with and understand. Analyzing the terms of an algebraic expression for common factors is the first and crucial step in the simplification process.

What are Algebraic Expressions?

Algebraic expressions are combinations of variables, numbers, and operations. They are the language of algebra and provide a way to describe mathematical situations. In the exercise example, \(x(2x+1) + 4(2x+1)\), we have two terms. Each term consists of numbers and variables combined through multiplication and addition.

To better understand these expressions, consider that they can be

• Monomials, like \(3x\), having only one term
• Binomials, like \(x + 4\), containing two terms
• Or even trinomials and polynomials, which include three or more terms

Algebraic expressions are the building blocks of equations, and knowing how they can be manipulated is key to mastering algebra. Recognizing patterns, such as common factors, helps in simplifying and solving these expressions.

Steps to Polynomial Simplification

Simplifying polynomials involves factoring them into a more easily managed form. The goal is to break down a complex expression into simpler parts. This not only makes it simpler to understand but also easier for further manipulation.

In simplifying the expression \(x(2x+1) + 4(2x+1)\):

• First, identify the common factor, which is \((2x + 1)\)
• Next, factor out the \((2x + 1)\) as a single term
• Rewrite the expression as \((2x + 1)(x + 4)\)

By performing these steps, we simplify the polynomial expression to a product of two factors. This simplification allows better interpretability and opens pathways for solving or further transforming the expression. The act of polynomial simplification is a vital skill in algebra, useful in solving equations, integration, and mathematical modeling.