Problem 55 - Elementary Algebra

Question

Factor out the GCF. $$ 2 x(x-3)+5(x-3) $$

Step-by-Step Solution

Answer

The expression factors to \((x-3)(2x+5)\).

Step 1: Identify Common Factor

First, identify the terms in the expression. The given expression is \(2x(x-3) + 5(x-3)\). Notice that \((x-3)\) is a common factor in both terms.

Step 2: Factor Out the Common Factor

Since \((x-3)\) is common in both \(2x(x-3)\) and \(5(x-3)\), you can factor it out. When you factor \((x-3)\) out, the expression becomes \((x-3)(2x + 5)\).

Step 3: Verify the Factored Expression

To ensure that the factored expression is correct, expand \((x-3)(2x+5)\) back to its original form. Distribute \((x-3)\) to both terms: \(2x(x-3) + 5(x-3)\). This confirms the factored expression is accurate.

Key Concepts

Greatest Common Factor (GCF)

The concept of the Greatest Common Factor, or GCF, is a key element when dealing with algebraic expressions. It represents the largest factor that two or more terms share. This can either be a number, a variable, or more commonly, a combination of both. Understanding how to find the GCF is essential when simplifying expressions or equations. To find the GCF in an algebraic expression, follow these steps:

• Identify each term in the expression separately.
• Examine both the numerical coefficients and the variables included in each term.
• Determine the factors common to all terms; this includes both numbers and variables present in each term. The lowest power of each variable that appears in all the terms should be used.

The GCF plays a crucial role during the factoring process as it allows the expression to be simplified, by "factoring out" the common element, making it easier to solve or manipulate further.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators such as addition, subtraction, multiplication, and division. In an expression like the one provided, there are several components you should consider:

• Terms: These are the individual parts of the expression separated by operators. For example, in the expression \(2x(x-3) + 5(x-3)\), there are two terms \(2x(x-3)\) and \(5(x-3)\).
• Coefficients: This is the numerical part of a term that is multiplied by the variable. In \(2x(x-3)\), the coefficient is 2.
• Variables: These represent unknown values and are usually denoted by letters such as \(x\). Variables allow expressions to be generalized and used in a variety of contexts.

Understanding the structure of algebraic expressions is fundamental to manipulating them correctly, including operations like factoring, expanding, and simplifying.

Factoring Techniques

Factoring is a valuable technique in algebra used to simplify expressions and solve equations. When factoring, you are essentially reversing the distributive property to express a sum or difference of terms as a product of simpler expressions.Here are some essential factoring techniques:

• Factoring out the GCF: This is often the first step in factoring and involves removing the greatest common factor from each term, as demonstrated in the given exercise.
• Grouping: Used when an expression involves four or more terms. Group terms with common factors, then factor out the GCF from each group.
• Special Patterns: Recognizing patterns like the difference of squares \((a^2 - b^2 = (a-b)(a+b))\) or trinomial squares can help factor expressions quickly.

In our exercise, factoring out the common factor \((x-3)\) simplifies the expression to \((x-3)(2x+5)\). Verifying the factored form by re-expanding it ensures accuracy. Mastery of factoring techniques opens the door to more complex algebraic manipulations and problem-solving.