Finding The Term For A Greatest Common Factor Of 12t³

Jul 15, 2025 by ADMIN 54 views

Which Term Can Be Added to Achieve a Greatest Common Factor of 12t³?

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In mathematics, finding the greatest common factor (GCF) is a fundamental concept, especially when dealing with algebraic expressions. The GCF, also known as the highest common factor (HCF), is the largest factor that divides two or more numbers or terms without leaving a remainder. Understanding how to determine the GCF is crucial for simplifying expressions, solving equations, and various other mathematical operations. In this article, we will explore how to identify the term that, when added to a list, results in a specific GCF for the entire set of terms. Let's dive into the concept of GCF and its application in the given problem.

Understanding the Greatest Common Factor (GCF)

To effectively tackle the problem of finding a term that results in a specific GCF, it's essential to have a solid understanding of what the GCF represents. The greatest common factor of a set of numbers or algebraic terms is the largest number or term that divides all the members of the set evenly. In simpler terms, it's the highest number that can be factored out from all the terms. When dealing with algebraic terms, the GCF includes both the numerical coefficients and the variable components.

The process of finding the GCF involves identifying the factors of each term and then selecting the common factors. The GCF is the product of the highest powers of these common factors. For example, to find the GCF of $12x^2$ and $18x^3$, we first identify the factors of the coefficients (12 and 18) and the variables ($x^2$ and $x^3$). 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, with 6 being the greatest. For the variables, $x^2$ is a factor of both $x^2$ and $x^3$, so it is the highest power of x that is common to both terms. Thus, the GCF of $12x^2$ and $18x^3$ is $6x^2$. Understanding this process is critical for solving problems involving GCFs and algebraic terms.

Moreover, it's important to note that the GCF can significantly simplify mathematical expressions and equations. By factoring out the GCF, we can reduce the complexity of the terms and make them easier to work with. This is particularly useful in algebra, where simplifying expressions is a common task. For instance, if we have an expression like $12x^2 + 18x^3$, factoring out the GCF, which we determined to be $6x^2$, gives us $6x^2(2 + 3x)$. This simplified form is not only easier to handle but also provides insights into the structure of the original expression. Therefore, mastering the concept of GCF is not just about finding the highest common factor but also about understanding its implications and applications in various mathematical contexts. The ability to quickly and accurately determine the GCF is a valuable skill that will aid in solving a wide range of problems.

Problem Statement: Finding the Missing Term

Our problem presents a specific challenge: we are given two terms, $36t^3$ and $12t^6$, and we need to find a third term that, when added to the list, makes the greatest common factor (GCF) of all three terms equal to $12t^3$. This type of problem requires a careful analysis of the given terms and the potential options to identify which one fits the criteria. The target GCF, $12t^3$, sets the standard we need to match. This means that each term in the final set must be divisible by $12t^3$, and no higher common factor should be present. The challenge lies in determining which of the provided options will satisfy this condition.

To approach this problem systematically, we must first break down the given terms and the potential options into their factors. This involves looking at both the numerical coefficients and the variable components. For the given terms, $36t^3$ can be factored as $12t^3 * 3$, and $12t^6$ can be factored as $12t^3 * t^3$. This immediately shows that $12t^3$ is a common factor of these two terms. Now, we need to examine the options and see which one, when combined with the original terms, keeps $12t^3$ as the GCF while preventing any higher common factors from emerging. This involves a process of elimination and verification.

Each option presents a different combination of numerical and variable factors. We need to ensure that the selected term is divisible by 12 and contains $t^3$ as the highest power of t that is common to all three terms. This requires a careful comparison of the exponents of t and the numerical coefficients. Additionally, it's crucial to consider what happens if we were to choose a term with a higher power of t or a coefficient that has common factors with 36 and 12, other than those already present in $12t^3$. Such a term might change the GCF to something other than $12t^3$. Therefore, solving this problem is not just about finding a common factor but about ensuring that the greatest common factor remains exactly as specified. The solution requires a precise understanding of GCF and careful consideration of how different terms interact in terms of their factors.

Analyzing the Options

Now, let's methodically examine each of the provided options to determine which one, when added to the list $36t^3$ and $12t^6$, results in a GCF of $12t^3$. This involves breaking down each option into its factors and comparing them with the factors of the given terms.

Option A: $6t^3$

The first option is $6t^3$. While $t^3$ is a factor of this term, and 6 is a factor of 12, the GCF of 36, 12, and 6 is 6, not 12. Therefore, the GCF of $36t^3$, $12t^6$, and $6t^3$ would be $6t^3$, not $12t^3$. Thus, option A does not meet the requirement of the problem.

Option B: $12t^2$

The second option is $12t^2$. Here, 12 is indeed a factor, but the power of t is $t^2$, which is less than $t^3$. The GCF of the variable parts will be the lowest power of t among the terms, which in this case would be $t^2$. Therefore, the GCF of $36t^3$, $12t^6$, and $12t^2$ would be $12t^2$, not $12t^3$. Option B is also incorrect.

Option C: $30t^4$

Option C presents $30t^4$. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. The common factors of 36, 12, and 30 are 1, 2, 3, and 6. This means the greatest common numerical factor would be 6, not 12. Additionally, the powers of t are $t^3$, $t^6$, and $t^4$. The lowest power of t is $t^3$. Thus, the GCF of $36t^3$, $12t^6$, and $30t^4$ would be $6t^3$, which again does not match the required GCF of $12t^3$. So, option C is not the correct answer.

Option D: $48t^5$

Finally, let’s consider option D, $48t^5$. The number 48 can be factored as $12 * 4$. Therefore, 12 is a factor of 48. The variable part is $t^5$. Now we need to check if $12t^3$ is the GCF of $36t^3$, $12t^6$, and $48t^5$. The GCF of the numerical coefficients 36, 12, and 48 is indeed 12. The powers of t are $t^3$, $t^6$, and $t^5$. The lowest power of t among these is $t^3$. Thus, the GCF of the three terms is $12t^3$, which matches the desired GCF.

By systematically analyzing each option, we've narrowed down the possibilities and identified the correct term that, when added to the list, results in the specified GCF. This methodical approach is crucial for solving problems involving GCFs and algebraic expressions.

Solution: Identifying the Correct Term

After a thorough analysis of each option, we have determined that Option D, $48t^5$, is the correct term to add to the list $36t^3$ and $12t^6$ so that the greatest common factor (GCF) of the three terms is $12t^3$. This conclusion is based on the following reasoning:

Numerical Coefficient: The coefficients of the three terms are 36, 12, and 48. The GCF of these numbers is 12, as 12 is the largest number that divides all three evenly (36 = 12 * 3, 12 = 12 * 1, 48 = 12 * 4).
Variable Component: The variable components are $t^3$, $t^6$, and $t^5$. The GCF of these variable terms is $t^3$, as it is the lowest power of t present in all three terms. The GCF is determined by the lowest exponent because this is the highest power of t that can divide all the terms without leaving a remainder.

Combining the numerical and variable GCFs, we get $12t^3$, which is the desired GCF. Adding $48t^5$ to the list ensures that this GCF is maintained. This is because $48t^5$ is divisible by $12t^3$ ( $48t^5 = 12t^3 * 4t^2$), and no higher common factor is introduced.

The other options were eliminated because they either resulted in a different numerical GCF (options A and C) or a different variable GCF (option B). Option A, $6t^3$, had a numerical GCF of 6, which is not 12. Option B, $12t^2$, had a variable GCF of $t^2$, which is less than $t^3$. Option C, $30t^4$, had a numerical GCF of 6 and a variable GCF of $t^3$, resulting in an overall GCF of $6t^3$, not $12t^3$.

Therefore, the systematic approach of breaking down each term into its factors and comparing them against the target GCF allowed us to confidently identify the correct answer. Understanding how numerical and variable factors combine to form the GCF is crucial for solving these types of problems effectively.

Conclusion

In summary, the problem required us to identify a term that, when added to the list $36t^3$ and $12t^6$, would result in a greatest common factor (GCF) of $12t^3$. Through a methodical analysis of each option, we determined that Option D, $48t^5$, is the correct answer. This conclusion was reached by examining the numerical and variable factors of each term and ensuring that the resulting GCF matched the specified value.

The process involved several key steps. First, we defined the concept of GCF and highlighted its importance in simplifying mathematical expressions. Then, we broke down the problem statement, emphasizing the need to find a term that fits the GCF criteria without introducing a higher common factor. Next, we systematically analyzed each option, comparing its factors with those of the given terms and the target GCF. Finally, we verified that $48t^5$ met the requirements by confirming that the GCF of 36, 12, and 48 is 12, and the GCF of $t^3$, $t^6$, and $t^5$ is $t^3$, thus resulting in a combined GCF of $12t^3$.

This problem underscores the importance of a solid understanding of GCF and its application in algebraic expressions. By mastering the process of identifying and comparing factors, one can effectively solve problems involving GCFs. The ability to break down complex terms into their components and analyze them methodically is a valuable skill in mathematics. Furthermore, this problem demonstrates how a step-by-step approach can lead to the correct solution, even when faced with multiple options. The key takeaway is that a clear understanding of the underlying mathematical concepts, combined with a systematic problem-solving strategy, is essential for success in mathematics and related fields.