Which Expression Has A Greatest Common Factor Of 3h A Step By Step Solution
In the realm of mathematics, the greatest common factor (GCF), also known as the highest common factor (HCF), is a foundational concept, especially when dealing with algebraic expressions. Understanding the GCF is essential for simplifying expressions, factoring polynomials, and solving various mathematical problems. In this article, we will dive deep into the concept of GCF and address the question: Which expression has a greatest common factor of 3h? We'll analyze several expressions and break down the process of identifying the GCF, making it clear and understandable for students and math enthusiasts alike.
The greatest common factor is the largest factor that two or more numbers (or algebraic terms) share. For instance, if we consider the numbers 12 and 18, the factors of 12 are 1, 2, 3, 4, 6, and 12, while the factors of 18 are 1, 2, 3, 6, 9, and 18. The common factors are 1, 2, 3, and 6, and the greatest among them is 6. Therefore, the GCF of 12 and 18 is 6. In algebraic expressions, we extend this concept to include variables and their powers. To find the GCF of algebraic expressions, we identify the common factors in both the coefficients (the numerical part) and the variables (the literal part). For example, to find the GCF of $12x^2$ and $18x$, we first find the GCF of the coefficients 12 and 18, which is 6. Then, we look at the variables. Both terms have $x$, but the lowest power of $x$ present in both terms is $x^1$ or simply $x$. Therefore, the GCF of $12x^2$ and $18x$ is $6x$. Mastering the technique of finding the greatest common factor is not just a mathematical exercise; it's a critical skill for simplifying complex expressions, solving equations, and understanding more advanced topics such as polynomial factorization. The GCF serves as the backbone for many algebraic manipulations and problem-solving strategies, making it an indispensable tool in the mathematician's toolkit.
To accurately answer the question of which expression has a greatest common factor of $3h$, we must thoroughly examine each option provided. This involves breaking down each expression into its constituent terms and identifying the common factors between them. The key here is to look at both the numerical coefficients and the variable components of each term.
Option A: $3 - 9h$
Let's start with the expression $3 - 9h$. This expression has two terms: 3 and $-9h$. The factors of 3 are 1 and 3. The factors of $-9h$ are -1, 1, -3, 3, -9, 9, -h, h, -3h, 3h, -9h, and 9h. The common factors between 3 and $-9h$ are 1, 3, -1, -3. Considering the greatest common factor, we focus on the largest numerical value, which is 3. There is no $h$ in the first term, so $h$ cannot be part of the GCF. Therefore, the GCF of $3 - 9h$ is 3.
Option B: $12h + 21h^2$
Next, we consider the expression $12h + 21h^2$. This expression also has two terms: $12h$ and $21h^2$. The factors of 12 are 1, 2, 3, 4, 6, and 12, while the factors of 21 are 1, 3, 7, and 21. The common factors between 12 and 21 are 1 and 3. So, the greatest common numerical factor is 3. Now, let's look at the variable part. The first term has $h$, and the second term has $h^2$. The lowest power of $h$ present in both terms is $h^1$, or simply $h$. Thus, the GCF of $12h$ and $21h^2$ is $3h$. This makes Option B a potential answer to our question.
Option C: $18h^2 - 6h^2$
The third expression is $18h^2 - 6h^2$. Before we find the GCF, it's important to note that this expression can be simplified by combining like terms. Subtracting $6h^2$ from $18h^2$ gives us $12h^2$. Since there is only one term after simplification, the GCF is simply the term itself, which is $12h^2$. Here, the numerical coefficient is 12, and the variable part is $h^2$. This means that the GCF is $12h^2$ and not $3h$.
Option D: $6h^2 - 2h$
Finally, let's analyze the expression $6h^2 - 2h$. The terms are $6h^2$ and $-2h$. The factors of 6 are 1, 2, 3, and 6, while the factors of -2 are -1, 1, -2, and 2. The common factors between 6 and -2 are 1 and 2, making 2 the greatest common numerical factor. Both terms contain $h$, with the lowest power of $h$ being $h^1$ or $h$. Therefore, the GCF of $6h^2$ and $-2h$ is $2h$.
To solve the problem "Which expression has a greatest common factor of 3h?", we need to systematically find the greatest common factor (GCF) of each provided expression. This involves breaking down each expression into its individual terms and identifying the factors that are common across all terms. The GCF is the product of the highest common numerical factor and the lowest power of common variables.
Step 1: Understanding the Question
The first step in solving any mathematical problem is to understand exactly what is being asked. In this case, we are looking for an algebraic expression where the greatest common factor (GCF) of its terms is $3h$. This means we need to find the expression where $3h$ is the largest factor that can divide evenly into each term.
Step 2: Analyzing Option A: $3 - 9h$
The expression is $3 - 9h$. Let’s break down each term:
- The first term, 3, has factors 1 and 3.
- The second term, $-9h$, has factors -1, 1, -3, 3, -9, 9, -h, h, -3h, 3h, -9h, and 9h.
The common factors are 1, 3, -1, and -3. The greatest common numerical factor is 3. However, the variable $h$ is only present in the second term, not in the first. Therefore, the GCF of this expression is 3, not $3h$.
Step 3: Analyzing Option B: $12h + 21h^2$
The expression is $12h + 21h^2$. Let’s break down each term:
- The first term, $12h$, has factors 1, 2, 3, 4, 6, 12, and $h$.
- The second term, $21h^2$, has factors 1, 3, 7, 21, $h$, and $h^2$.
The common numerical factors of 12 and 21 are 1 and 3. The greatest common numerical factor is 3. Both terms have the variable $h$. The lowest power of $h$ in both terms is $h^1$, or simply $h$. Therefore, the GCF of $12h$ and $21h^2$ is $3h$.
Step 4: Analyzing Option C: $18h^2 - 6h^2$
The expression is $18h^2 - 6h^2$. Before finding the GCF, we can simplify the expression by combining like terms:
Since there is only one term, the GCF is the term itself, which is $12h^2$. This GCF is not $3h$.
Step 5: Analyzing Option D: $6h^2 - 2h$
The expression is $6h^2 - 2h$. Let’s break down each term:
- The first term, $6h^2$, has factors 1, 2, 3, 6, $h$, and $h^2$.
- The second term, $-2h$, has factors -1, 1, -2, 2, and $h$.
The common numerical factors of 6 and -2 are 1 and 2. The greatest common numerical factor is 2. Both terms have the variable $h$. The lowest power of $h$ in both terms is $h^1$, or simply $h$. Therefore, the GCF of $6h^2$ and $-2h$ is $2h$, which is not $3h$.
Step 6: Conclusion
After analyzing each option, we found that the expression $12h + 21h^2$ has a greatest common factor of $3h$. Therefore, the correct answer is Option B.
After a comprehensive analysis of each expression, it is evident that Option B, $12h + 21h^2$, is the correct answer. The step-by-step solution demonstrated how to identify the greatest common factor by breaking down each term and finding the common factors. The GCF of $12h + 21h^2$ is indeed $3h$, as it is the largest factor that divides evenly into both $12h$ and $21h^2$. Options A, C, and D were found to have GCFs of 3, $12h^2$, and $2h$, respectively.
Understanding GCF is not just about finding a common factor; it's about finding the greatest common factor. This concept is crucial in simplifying expressions, solving equations, and understanding more advanced algebraic concepts. By mastering the technique of identifying the GCF, students can develop a stronger foundation in mathematics and improve their problem-solving skills.
In conclusion, when faced with problems involving greatest common factors, it is essential to break down each term, identify common numerical and variable factors, and systematically determine the largest common factor. This approach ensures accuracy and builds a deeper understanding of mathematical principles. The correct answer to the question "Which expression has a greatest common factor of $3h$?" is B. $12h + 21h^2$.
