Problem 8 - Elementary Algebra

Question

List, if any should appear, the common factors for the following problems. $$ 6(a+4)+12(a+4) $$

Step-by-Step Solution

Answer

Answer: The common factors in the given expression are 6.

Step 1: 1. Simplify the expression

First, we need to simplify the given expression: $$ 6(a+4)+12(a+4) $$ We can distribute the terms inside the parentheses to obtain: $$ 6a+24+12a+48 $$

Step 2: 2. Combine like terms

Next, we can combine the like terms (6a and 12a, as well as 24 and 48) to get a simplified expression: $$ (6a+12a)+(24+48) $$ $$ 18a+72 $$

Step 3: 3. Find common factors

Now, we can identify the common factors present in both terms of the simplified expression (18a and 72). The common factors are the numbers or variables that can divide both terms without leaving a remainder. In this case, we can see that the common factor is 6: $$ 6(3a+12) $$ So, the common factors in the given expression are 6.

Key Concepts

Factors in Algebra

When working with algebraic expressions, identifying factors is crucial. Factors are elements that multiply to form a product. In algebra, we focus on numbers, variables, or combinations that multiply to form another value. For instance, in the expression \(18a + 72\), we search for common factors in both terms.

Factors can be both numerical and algebraic. To find factors, we often perform factorization, where we break down a complex expression into simpler multiplicative parts.

Consider these steps for finding factors:

• List all possible divisors of the numerical coefficients.
• Search for common variables present in all terms.
• Identify the greatest common factor (GCF) that divides all terms without a remainder.

By identifying and extracting factors, we simplify expressions, making them easier to solve, calculate, or further analyze.

Like Terms in Algebra

In algebra, recognizing like terms is essential for simplifying expressions. Like terms are terms that have identical variable parts, allowing them to be combined through addition or subtraction. They make arithmetic operations straightforward and help condense expressions.

For example, in the expression \(6a + 12a\), both terms are like terms because "\(a\)" is the variable in both. You can combine them by adding their coefficients together, yielding \(18a\).

To master like terms, follow these simple guidelines:

• Identify terms with the same variables and exponents.
• Combine them by adding or subtracting their coefficients.
• Be cautious of signs (positive/negative) when combining terms.

By correctly combining like terms, expressions are not only simplified but also become more manageable for further algebraic operations.

Distributive Property in Algebra

The distributive property is a fundamental algebraic principle that allows simplification of expressions by multiplying a single term across terms in parentheses. It is expressed as \(a(b + c) = ab + ac\). This property ensures every term inside the parentheses is multiplied by the term outside.

Let's look at the expression \(6(a+4) + 12(a+4)\). Using the distributive property, each term within the parentheses, \(a+4\), is multiplied by both 6 and 12. This process converts the expression to \(6a + 24 + 12a + 48\).

The distributive property facilitates:

• Expansion of expressions for simplification.
• Simplifying complex equations or expressions.
• Setting the stage for further operations such as combining like terms or finding factors.

Understanding and applying the distributive property efficiently can simplify your calculations and make algebraic manipulations more intuitive.