GCF: Finding The Missing Term For 12h³

Let's dive into a fun math problem where we need to find the missing piece to make a puzzle complete! In this case, our puzzle involves finding the greatest common factor (GCF) of algebraic terms. Think of the GCF as the largest factor that can divide evenly into all the terms we're considering. Our mission, should we choose to accept it, is to figure out which term we can add to a list so that the GCF of all the terms becomes a specific value. Buckle up, because we're about to embark on a mathematical adventure!

Understanding the Greatest Common Factor (GCF)

Before we jump into the problem, let's make sure we're all on the same page about what the greatest common factor actually is. The GCF, as we mentioned earlier, is the largest factor that divides evenly into two or more numbers or terms. When we're dealing with numbers, it's pretty straightforward. For example, the GCF of 12 and 18 is 6, because 6 is the biggest number that divides both 12 and 18 without leaving a remainder.

But what about algebraic terms, like the ones in our problem? Well, the concept is the same, but we also need to consider the variables and their exponents. To find the GCF of algebraic terms, we look for the largest numerical factor that divides all the coefficients (the numbers in front of the variables) and the lowest power of each variable that appears in all the terms. For instance, if we have the terms $x^3$ and $x^5$, the lowest power of x that appears in both is $x^3$. So, $x^3$ would be part of the GCF.

To truly master GCF, let’s break down the process into simple steps. First, identify the coefficients in each term. These are the numerical parts of the terms. Next, find the greatest common factor of these coefficients. This is the largest number that divides evenly into all the coefficients. Then, look at the variables in each term. For each variable, find the lowest exponent that appears across all terms. This lowest exponent will be the exponent of that variable in the GCF. Finally, combine the GCF of the coefficients with the variables raised to their lowest exponents to get the overall GCF of the terms. Remember, the GCF is like the foundation upon which all the terms are built, so understanding it is crucial for solving this type of problem.

The Problem at Hand

Now that we've refreshed our understanding of the GCF, let's take a closer look at the problem we're trying to solve. We're given a list of two terms: $36h^3$ and $12h^6$. Our goal is to find a third term that we can add to this list so that the GCF of all three terms is $12h^3$. We're also given four options to choose from:

• A. $6h^3$
• B. $12h^2$
• C. $30h^4$
• D. $48h^5$

To tackle this problem, we need to carefully analyze each option and see if adding it to the list would result in a GCF of $12h^3$. We'll need to consider both the numerical coefficients and the powers of the variable 'h'. It might seem a bit daunting at first, but don't worry! We'll break it down step by step and make sure we understand the reasoning behind each step.

Remember, the GCF must divide evenly into all terms, so we need to make sure that the numerical coefficient in our chosen term is divisible by 12, and the power of 'h' in our chosen term must be at least $h^3$ (since that's the power of 'h' in the desired GCF). Keep these key points in mind as we evaluate each option.

Analyzing the Options

Let's go through each option one by one and see if it fits the bill. We'll start with option A, $6h^3$, and then move on to the others. For each option, we'll determine the GCF of the three terms (the two original terms plus the option) and see if it matches our target GCF of $12h^3$.

Option A: $6h^3$

If we add $6h^3$ to our list, we have the terms $36h^3$, $12h^6$, and $6h^3$. To find the GCF, we first look at the coefficients: 36, 12, and 6. The GCF of these numbers is 6 (since 6 is the largest number that divides all three). Next, we look at the powers of 'h'. We have $h^3$, $h^6$, and $h^3$. The lowest power of 'h' is $h^3$. So, the GCF of the three terms would be $6h^3$. This does not match our target GCF of $12h^3$, so option A is not the correct answer.

Option B: $12h^2$

Now let's consider option B, $12h^2$. Adding this to our list gives us the terms $36h^3$, $12h^6$, and $12h^2$. Again, we start by looking at the coefficients: 36, 12, and 12. The GCF of these numbers is 12. Now, let's look at the powers of 'h': $h^3$, $h^6$, and $h^2$. The lowest power of 'h' is $h^2$. Therefore, the GCF of the three terms would be $12h^2$. This also does not match our target GCF of $12h^3$, so option B is incorrect.

Option C: $30h^4$

Moving on to option C, $30h^4$, our list of terms becomes $36h^3$, $12h^6$, and $30h^4$. The coefficients are 36, 12, and 30. The GCF of these numbers is 6 (the largest number that divides 36, 12, and 30). The powers of 'h' are $h^3$, $h^6$, and $h^4$. The lowest power of 'h' is $h^3$. Thus, the GCF of the three terms is $6h^3$. This is not equal to our desired GCF of $12h^3$, so option C is not the right answer.

Option D: $48h^5$

Finally, let's analyze option D, $48h^5$. Our list of terms now includes $36h^3$, $12h^6$, and $48h^5$. The coefficients are 36, 12, and 48. The GCF of these numbers is 12 (since 12 is the largest number that divides all three). The powers of 'h' are $h^3$, $h^6$, and $h^5$. The lowest power of 'h' is $h^3$. Therefore, the GCF of the three terms is $12h^3$. This does match our target GCF! So, option D is the correct answer.

The Solution

After carefully analyzing each option, we've found that the term we can add to the list $36h^3$, $12h^6$ so that the greatest common factor of the three terms is $12h^3$ is D. $48h^5$. We arrived at this solution by systematically determining the GCF of the three terms for each option and comparing it to our target GCF. By breaking down the problem into smaller, manageable steps, we were able to confidently identify the correct answer.

Key Takeaways

This problem highlights the importance of understanding the concept of the greatest common factor, especially when dealing with algebraic terms. Remember, the GCF is the largest factor that divides evenly into all the terms, and it includes both the numerical coefficients and the variables raised to their lowest powers. By carefully analyzing each term and systematically determining the GCF, we can solve problems like this with ease. This skill is not just useful for math problems; it also helps in simplifying expressions and understanding relationships between different quantities in various fields.

Further Exploration

Want to delve deeper into the world of GCF and LCM? Check out this helpful resource on Khan Academy. It's a fantastic place to practice your skills and explore more challenging problems. Happy calculating!