Problem 8 - College Algebra Essentials

Question

$$\text { Factor out the greatest common factor.}$$ $$x(2 x+1)+4(2 x+1)$$

Step-by-Step Solution

Answer

The factored expression of \(x(2x+1)+4(2x+1)\) is \((2x+1)(x+4)\)

Step 1: Identify the Common Factor

The expression is \(x(2x+1)+4(2x+1)\). There are two terms in the expression: \(x(2x+1)\) and \(4(2x+1)\). The common factor between these terms is \((2x+1)\).

Step 2: Factor out the Common Factor

Once we have identified the common factor \((2x+1)\), we can factor it out from each term. We get \((2x+1)\) to be factored out, leaving \(x+4\) inside the parentheses. Thus, the expression will become: \((2x+1)(x+4)\).

Key Concepts

Greatest Common Factor

The greatest common factor, often abbreviated as GCF, is a key concept in algebra that helps simplify expressions and solve equations. It is the largest number or expression that can exactly divide each term of the given algebraic expression without leaving a remainder.

When factoring algebraic expressions, identifying the GCF is the first and crucial step. For example, in the expression given from the exercise, each term, specifically \( x(2x+1) \) and \( 4(2x+1) \), contains the common factor \((2x+1)\). This factor is shared and can be taken out of the expression, leading to a simpler form.

Determining the greatest common factor involves:

• Identifying the common factors in all terms.
• Selecting the highest degree of any common literal factors.
• Factoring it out, simplifying the expression significantly.

Once factored, the expression becomes easier to work with or solve.

Algebraic Expressions

Algebraic expressions play a vital role in mathematics and are central to algebra. They consist of variables, constants, and arithmetic operations (like addition, subtraction, multiplication, and division). These expressions are formulas that represent mathematical concepts in a concise form.

Consider an algebraic expression as a way to express computations or values in terms of symbols with no equal sign present. For example, in the expression from the exercise, \(x(2x+1)+4(2x+1)\) is an algebraic expression with two terms.

Key features of algebraic expressions include:

• Variables: Often represented by letters (e.g., \(x, y\)) that stand for number values.
• Constants: These are fixed numerical values (e.g., \(2, 4\)).
• Operations: The specific arithmetic actions applied to variables and constants (e.g., addition with \((+)\) or multiplication with \(()\)).

Understanding how to manipulate these expressions, such as factoring them, forms the basis of solving algebraic equations and inequalities.

Polynomials

Polynomials are a specific type of algebraic expression essential in various mathematical problems and real-world applications. These expressions include terms constructed using coefficients, variables, and exponents, where the exponents are non-negative integers.

The expression \(x(2x+1)+4(2x+1)\) from the exercise demonstrates the structure of a polynomial. Each term, such as \(x(2x+1)\), within the expression can include coefficients (numbers preceding variables) and variables (\(x\) here used) raised to a power.

Some distinct properties of polynomials are:

• Terms: Made up of combinations of variables and coefficients.
• Degree: Defined by the highest exponent in the polynomial (e.g., the degree of \(2x^2+3x+4\) is 2).
• Operations: May involve addition, subtraction, and multiplication but not division by variables.

Understanding polynomials and their properties is crucial for solving equations, performing algebraic operations, and is foundational for advanced topics in mathematics.