Problem 22 - Algebra for College Students

Question

Factor completely. $$12 x+8 y$$

Step-by-Step Solution

Answer

The expression factors to \(4(3x + 2y)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, look at the terms in the expression \(12x + 8y\). You need to find the greatest common factor of the coefficients 12 and 8, which are the numbers in front of the variables \(x\) and \(y\). Since both 12 and 8 are divisible by 4, the GCF is 4.

Step 2: Factor out the GCF

Now, divide each term in the original expression \(12x + 8y\) by the GCF which is 4. This means you divide 12x and 8y separately: \(12x \div 4 = 3x\) and \(8y \div 4 = 2y\). Thus, when you factor out 4, you get \(4(3x + 2y)\).

Step 3: Verify the Factored Expression

Check your work by distributing the GCF back to each term inside the parentheses. Multiply 4 by each term: \(4 \times 3x = 12x\) and \(4 \times 2y = 8y\). The result should equal the original expression, \(12x + 8y\), confirming that \(4(3x + 2y)\) is indeed the correct factorization.

Key Concepts

Greatest Common Factor

The greatest common factor, or GCF, is the largest number that divides two or more numbers without leaving a remainder. Think of it as the biggest building block shared by different numbers, which makes it quite powerful in simplifying expressions. To find the GCF, you should:

• List the factors of each number involved in the expression.
• Identify the largest factor that appears in every list.

Let's consider the expression with the coefficients 12 and 8 in our exercise. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 8 are 1, 2, 4, and 8. The largest common factor is 4. Hence, the GCF is 4, and this is what you'll "factor out" to simplify the expression. Understanding GCF is crucial for more efficient calculations and also in reducing fractions.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations like addition or multiplication. It is a mathematical phrase that can represent real-world situations and various quantities. In the exercise, the expression is given as:\[12x + 8y\]Here, the numbers 12 and 8 are coefficients, and \(x\) and \(y\) are variables that can take any value. The goal is to make such expressions simpler to work with, especially when solving equations or simplifying expressions involving these terms. By factoring, we essentially "break down" the expression into simpler parts. This helps not only in simplification but also in solving equations where you need isolated variable terms. Recognizing and working with these components effectively is a fundamental skill in algebra.

Distributive Property

The distributive property is a useful tool in algebra that allows you to multiply a single term by each term within a set of parentheses. It's an essential property for manipulating algebraic expressions.Simply put, the distributive property states that:\[a(b + c) = ab + ac\]In our exercise, after factoring the expression \[12x + 8y\]and writing it as \[4(3x + 2y)\]you can check your factoring by distributing the 4 back to each term inside the parentheses:

• \(4 \times 3x = 12x\)
• \(4 \times 2y = 8y\)

This process should return you to the original expression, thereby confirming that the factorization is correct.Grasping the distributive property helps in breaking down and distributing operations across terms, providing clarity and simplification, greatly easing the task of handling more complex algebraic expressions.