Expression With Greatest Common Factor Of 3h

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In the realm of mathematics, understanding the greatest common factor (GCF) is crucial, especially when dealing with algebraic expressions. The greatest common factor, often abbreviated as GCF, is the largest factor that divides two or more numbers or algebraic terms without leaving a remainder. For expressions, the GCF includes both the largest numerical factor and the highest power of the variable that is common to all terms. This article delves into the process of identifying the expression that has a GCF of 3h. We will explore various expressions, break down their factors, and determine which one fits the criterion. This skill is not only fundamental in algebra but also plays a significant role in simplifying expressions, solving equations, and understanding polynomial manipulations. Let’s embark on this mathematical journey to master the concept of GCF and its application in algebraic expressions.

Understanding Greatest Common Factor (GCF)

Before we dive into the specific expressions, let's solidify our understanding of the greatest common factor (GCF). The GCF, in simple terms, is the largest number or expression that can divide evenly into a set of numbers or terms. For instance, if we consider the numbers 12 and 18, their factors are:

  • Factors of 12: 1, 2, 3, 4, 6, 12
  • Factors of 18: 1, 2, 3, 6, 9, 18

From the lists above, it is clear that the common factors are 1, 2, 3, and 6. The greatest among these common factors is 6. Therefore, the GCF of 12 and 18 is 6. This concept extends to algebraic expressions, where we not only look for the largest numerical factor but also the highest power of the variable that is common to all terms. For example, in the expression 4x² + 8x, the numerical factors of 4 and 8 are 1, 2, and 4. The highest power of x common to both terms is x. Thus, the GCF of 4x² and 8x is 4x.

Understanding how to find the GCF is essential for simplifying algebraic expressions and solving equations. It helps in reducing fractions to their simplest form, factoring polynomials, and various other algebraic manipulations. The process typically involves identifying the factors of each term and then picking out the largest factor they have in common. Let's illustrate this with a more complex example: Consider the expression 15a³b² + 25a²b³. The numerical factors of 15 and 25 are 1, 5. The highest power of 'a' common to both terms is a², and the highest power of 'b' common to both terms is b². Hence, the GCF of 15a³b² and 25a²b³ is 5a²b². Mastering this concept is crucial for more advanced topics in algebra, making it a cornerstone of mathematical proficiency.

Analyzing the Given Expressions

To determine which expression has a greatest common factor (GCF) of 3h, we must meticulously analyze each option. The expressions presented are:

  • A. 6h² - 2h
  • B. 3 - 9h
  • C. 12h + 21h²
  • D. 18h² - 6h

Let's break down each expression to identify their respective GCFs. This involves finding the largest numerical factor and the highest power of the variable 'h' that divides each term evenly. Understanding the structure of each term and how they relate to each other is key in this process. We will look at the coefficients (the numerical parts) and the variables separately to identify common elements. This methodical approach will ensure that we accurately determine the GCF for each expression.

Expression A: 6h² - 2h

In the expression 6h² - 2h, we identify two terms: 6h² and -2h. To find the greatest common factor (GCF), we examine the numerical coefficients and the variable parts separately. The numerical coefficients are 6 and -2. The factors of 6 are 1, 2, 3, and 6, while the factors of -2 are -1, -2, 1, and 2. The largest numerical factor they share is 2. Now, let's look at the variable part. We have h² in the first term and h in the second term. The highest power of 'h' that is common to both terms is h (since h² is h * h, and h is present in both terms). Combining the largest numerical factor and the highest power of 'h', we find that the GCF of 6h² and -2h is 2h. Therefore, expression A has a GCF of 2h.

Expression B: 3 - 9h

For the expression 3 - 9h, we again have two terms: 3 and -9h. The greatest common factor (GCF) here can be found by looking at the numerical parts and the variable parts. The numerical coefficients are 3 and -9. The factors of 3 are 1 and 3, while the factors of -9 are -1, -3, -9, 1, 3, and 9. The largest numerical factor they share is 3. Now, let's consider the variable part. The first term, 3, has no variable. The second term has 'h'. Since there is no 'h' in the first term, the variable part of the GCF is 1 (or essentially, no variable). Thus, the GCF of 3 and -9h is simply 3. Expression B has a GCF of 3.

Expression C: 12h + 21h²

Now, let's analyze the expression 12h + 21h². We have two terms: 12h and 21h². To determine the greatest common factor (GCF), we will examine the numerical coefficients and the variable parts separately. The numerical coefficients are 12 and 21. The factors of 12 are 1, 2, 3, 4, 6, and 12, while the factors of 21 are 1, 3, 7, and 21. The largest numerical factor common to both 12 and 21 is 3. Next, we consider the variable part. The first term has 'h', and the second term has h². The highest power of 'h' that is common to both terms is h. Therefore, the GCF of 12h and 21h² is the combination of the largest numerical factor and the highest common power of 'h', which is 3h. Hence, expression C has a GCF of 3h.

Expression D: 18h² - 6h

Finally, let's consider the expression 18h² - 6h. This expression has two terms: 18h² and -6h. We need to find the greatest common factor (GCF) by analyzing the numerical coefficients and the variable parts independently. The numerical coefficients are 18 and -6. The factors of 18 are 1, 2, 3, 6, 9, and 18, while the factors of -6 are -1, -2, -3, -6, 1, 2, 3, and 6. The largest numerical factor that both 18 and -6 share is 6. Looking at the variable part, we have h² in the first term and h in the second term. The highest power of 'h' that is common to both terms is h. Thus, the GCF of 18h² and -6h is 6h. Expression D has a GCF of 6h.

Selecting the Correct Answer

After meticulously analyzing each expression, we have determined the greatest common factor (GCF) for each:

  • A. 6h² - 2h has a GCF of 2h.
  • B. 3 - 9h has a GCF of 3.
  • C. 12h + 21h² has a GCF of 3h.
  • D. 18h² - 6h has a GCF of 6h.

The question asks us to identify the expression with a GCF of 3h. Based on our analysis, expression C, which is 12h + 21h², has a GCF of 3h. Therefore, the correct answer is C.

Conclusion

In conclusion, the expression that has a greatest common factor (GCF) of 3h is C. 12h + 21h². This determination was made through a detailed analysis of each expression, where we identified the largest numerical factor and the highest power of the variable 'h' common to all terms in the expression. Understanding the concept of GCF is essential in algebra for simplifying expressions and solving equations. By mastering this concept, one can tackle more complex algebraic problems with confidence and precision. The process of finding the GCF involves breaking down each term into its factors and then identifying the largest factor that is common among them. This skill is not only crucial for academic success in mathematics but also has practical applications in various fields that require mathematical proficiency.