Factoring: GCF Of 48y + 36z Explained!

by Alex Johnson 39 views

Let's break down how to factor the expression 48y + 36z using the greatest common factor (GCF). This is a fundamental concept in algebra, and understanding it will help you simplify more complex expressions later on. We'll go through the steps, explain the logic, and identify the correct answer from the given options.

Understanding the Greatest Common Factor (GCF)

In the realm of mathematics, particularly when dealing with algebraic expressions, the greatest common factor (GCF) stands as a pivotal concept. To truly grasp its significance, let's delve into a comprehensive exploration. The GCF, at its core, represents the largest number that can evenly divide into two or more numbers. When extending this concept to algebraic terms, the GCF encompasses not only the largest numerical factor but also the highest power of any variables common to those terms. Identifying the GCF serves as the initial yet crucial step in simplifying expressions, making it more manageable and easier to work with. This process is foundational in various mathematical operations such as solving equations, simplifying fractions, and comprehending the underlying structure of algebraic relationships. Mastering the art of finding the GCF empowers individuals to manipulate mathematical expressions with increased precision and efficiency, thereby laying a solid groundwork for more advanced mathematical pursuits.

Finding the GCF of 48 and 36

Our main goal here is to identify the largest number that can divide both 48 and 36 without leaving a remainder. Here's how we can do it:

  1. List the factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
  2. List the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
  3. Identify the common factors: 1, 2, 3, 4, 6, 12
  4. Determine the greatest common factor: From the list of common factors, the largest number is 12. Therefore, the GCF of 48 and 36 is 12.

Factoring Out the GCF from the Expression

Now that we've determined that the GCF of 48 and 36 is 12, we can proceed to factor it out of the expression 48y + 36z. This involves dividing each term in the expression by the GCF and rewriting the expression in a factored form. Factoring out the GCF is a fundamental technique in algebra used to simplify expressions and solve equations. It allows us to rewrite an expression as a product of the GCF and a simplified expression within parentheses. This process not only makes the expression more manageable but also reveals important relationships between its terms. Moreover, factoring out the GCF is a crucial step in various algebraic manipulations, such as solving quadratic equations, simplifying rational expressions, and identifying common factors in more complex expressions. By mastering this technique, one gains a valuable tool for tackling a wide range of algebraic problems with greater efficiency and accuracy. Understanding how to factor out the GCF is essential for success in algebra and beyond.

Factoring 48y + 36z

To factor the expression 48y + 36z, we need to divide each term by the GCF, which we found to be 12:

  • 48y ÷ 12 = 4y
  • 36z ÷ 12 = 3z

Now, we rewrite the expression using the GCF and the results of the division:

48y + 36z = 12(4y + 3z)

Analyzing the Given Options

We are given four options, and we need to determine which one correctly represents the factored form of 48y + 36z.

(A) 6(8y + 6z) (B) 16(3y + 2z) (C) 12(4y + 3z) (D) 4(12y + 9z)

To find the correct option, we can distribute the number outside the parenthesis into the terms inside the parenthesis, and see if we can obtain the original expression, 48y + 36z.

  • (A) 6(8y + 6z) = 48y + 36z
  • (B) 16(3y + 2z) = 48y + 32z
  • (C) 12(4y + 3z) = 48y + 36z
  • (D) 4(12y + 9z) = 48y + 36z

Evaluating Each Option

Now, let's individually evaluate each option to see which one matches our factored expression:

  • (A) 6(8y + 6z): Distributing the 6, we get 6 * 8y + 6 * 6z = 48y + 36z. While this equals the original expression, 6 is not the GCF of 48 and 36. Therefore, this option, while resulting in the original expression, is not fully factored using the greatest common factor.
  • (B) 16(3y + 2z): Distributing the 16, we get 16 * 3y + 16 * 2z = 48y + 32z. This does not equal the original expression. So, this option is incorrect.
  • (C) 12(4y + 3z): Distributing the 12, we get 12 * 4y + 12 * 3z = 48y + 36z. This equals the original expression, and 12 is the GCF of 48 and 36. Therefore, this is the correct option.
  • (D) 4(12y + 9z): Distributing the 4, we get 4 * 12y + 4 * 9z = 48y + 36z. While this equals the original expression, 4 is not the GCF of 48 and 36. Therefore, this option, while resulting in the original expression, is not fully factored using the greatest common factor.

Conclusion

After analyzing all the options, we found that only option (C) correctly factors the expression 48y + 36z using the greatest common factor (GCF), which is 12. Factoring out the GCF simplifies the expression and makes it easier to work with in various mathematical contexts. Therefore, the correct answer is:

(C) 12(4y + 3z)

Understanding how to find and use the greatest common factor is a crucial skill in algebra, allowing you to simplify expressions and solve equations more efficiently. Keep practicing, and you'll master this technique in no time!

For further learning, check out this resource on Greatest Common Factor (GCF) at Khan Academy