Problem 9 - Elementary Algebra

Question

List, if any should appear, the common factors for the following problems. $$ 4(a+2 b)+6(a+2 b) $$

Step-by-Step Solution

Answer

Answer: The common factors are 1, 2, 5, and 10, and the simplified expression is $$10(a+2b)$$.

Step 1: Distribute the numbers

First, we'll distribute the numbers and simplify the terms in the expression. The expression given is: $$ 4(a+2 b)+6(a+2 b) $$ We'll start by distributing the coefficients (4 and 6) to the terms inside the parenthesis: $$ 4a + 8b + 6a + 12b $$

Step 2: Combine like terms

Now, we'll combine the like terms to make the expression simpler: $$ (4a + 6a) + (8b + 12b) $$ This simplifies to: $$ 10a + 20b $$

Step 3: Find common factors of the simplified expression

Now, we need to find the common factors of the simplified expression: $$ 10a + 20b $$ To do this, we'll look at the coefficients of both terms (10 and 20) and identify which numbers can be evenly divided into both coefficients. 10 has factors of: 1, 2, 5, 10 20 has factors of: 1, 2, 4, 5, 10, 20 The common factors of 10 and 20 are: 1, 2, 5, 10

Step 4: Factor the common factors from the expression

Now that we have identified the common factors (1, 2, 5, 10), we will choose the largest common factor (10) and factor it out of the expression: $$ 10(1a + 2b) $$ The common factors for the given exercise are: 1, 2, 5, and 10. The simplified expression after factoring out the largest common factor (10) is: $$ 10(a+2b) $$

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to simplify and solve equations efficiently. It involves multiplying a single term by each term inside a parenthesis.

For example, in the expression \(4(a+2b)+6(a+2b)\), you apply the distributive property by multiplying 4 with \(a\) and \(2b\), and then the same with the 6. This is the first step towards simplifying complex algebraic expressions, enabling a clearer path to finding solutions. Though it might seem straightforward, making correct use of the distributive property is essential in avoiding mistakes while factoring and simplifying expressions.

Combining Like Terms

Combining like terms is a technique used to simplify an algebraic expression. Like terms are terms that have the same variables raised to the same power. For example, \(4a\) and \(6a\) are like terms because they both have the variable \(a\) raised to the same power.

In the expression \(4a + 6a + 8b + 12b\), you combine like terms to get \(10a + 20b\). This process of combining similar terms not only makes the expression shorter and easier to work with but also sets the stage for further simplification, such as factoring out common factors.

Finding Common Factors

Finding common factors involves identifying numbers that divide evenly into each of the numbers you are considering. Common factors are useful when you need to factor an algebraic expression or solve equations.

When examining the expression \(10a + 20b\), the common factors between the coefficients 10 and 20 are identified as 1, 2, 5, and 10. These numbers can all evenly divide both 10 and 20 without leaving a remainder. Recognizing these common factors is crucial as it enables you to factor expressions effectively, leading to simpler or more elegant solutions.

Simplifying Algebraic Expressions

Simplifying algebraic expressions means reducing them to their simplest form while retaining their value. This often includes distributing numbers, combining like terms, and factoring out common factors.

For instance, the original expression \(4(a+2b)+6(a+2b)\) is simplified through a series of steps, ultimately resulting in \(10(a+2b)\). This process facilitates a better understanding of the expression and lays the groundwork for solving complex problems. Simplification is a powerful tool in algebra that, with practice, becomes an invaluable skill for any student to master.