Determining The Term For A GCF Of 12t³
#h1 Introduction
In this article, we will delve into the process of identifying a term that, when added to the existing list of terms, results in a greatest common factor (GCF) of 12t³. Understanding the concept of GCF is crucial in various mathematical contexts, including simplifying expressions, solving equations, and working with polynomials. Our focus will be on determining the correct term from the given options that satisfies the GCF condition. We will explore the given terms, analyze their factors, and evaluate each option to arrive at the solution. The goal is to provide a comprehensive explanation that clarifies the concept and the method to solve this type of problem effectively.
#h2 Understanding Greatest Common Factor (GCF)
Before diving into the problem, let's define the greatest common factor (GCF). The GCF, also known as the greatest common divisor (GCD), is the largest number that divides two or more numbers without leaving a remainder. In the context of algebraic terms, the GCF includes both the numerical coefficient and the variable part. For instance, when finding the GCF of algebraic expressions, we look for the highest number that divides all coefficients and the highest power of the variable that is common to all terms. Identifying the GCF involves breaking down each term into its prime factors and then identifying the common factors raised to the lowest power present in any of the terms. This concept is fundamental in simplifying algebraic fractions and solving various algebraic problems. In our specific scenario, we are looking for a term that, when added to the existing terms, will yield a GCF of 12t³. This means the new term must be divisible by 12t³, and no higher factor can divide all three terms.
#h2 Problem Statement
We are given two terms: 36t³ and 12t⁶. The task is to find a third term from the options below that, when combined with the given terms, yields a greatest common factor (GCF) of 12t³.
The options are:
A. 6t³ B. 12t² C. 30t⁴ D. 48t⁵
To solve this, we must first determine the GCF of the given terms and then evaluate each option to see which one, when included, maintains the GCF at 12t³. This involves understanding how to find the GCF of algebraic terms, which includes identifying the greatest common numerical factor and the lowest power of the common variable. The correct option will be the one that does not introduce any new factors (either numerical or variable) that would change the GCF to something other than 12t³.
#h2 Finding the GCF of the Given Terms
To begin, let's find the greatest common factor (GCF) of the given terms: 36t³ and 12t⁶. The first step is to find the GCF of the coefficients, which are 36 and 12. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36, while the factors of 12 are 1, 2, 3, 4, 6, and 12. The greatest common factor of 36 and 12 is 12.
Next, we consider the variable part. We have t³ and t⁶. The GCF of variable terms is the variable raised to the lowest power present in the terms. In this case, the lowest power of t is 3, so the GCF of t³ and t⁶ is t³.
Combining these, the GCF of 36t³ and 12t⁶ is 12t³. This means that any term we add to the list must not change this GCF. It either needs to be a multiple of 12t³ or have factors that, when combined with the existing terms, still result in 12t³ as the GCF. This understanding is crucial for evaluating the options effectively.
#h2 Evaluating the Options
Now, let's evaluate each option to determine which one, when added to the list, maintains the greatest common factor (GCF) at 12t³.
Option A: 6t³
If we add 6t³ to the list, we have the terms 36t³, 12t⁶, and 6t³. The coefficients are 36, 12, and 6. The GCF of these numbers is 6. The variable part is t³, t⁶, and t³. The GCF of these is t³. Thus, the GCF of the three terms would be 6t³, which is not 12t³. Therefore, option A is incorrect.
Option B: 12t²
If we add 12t² to the list, we have 36t³, 12t⁶, and 12t². The coefficients are 36, 12, and 12, and their GCF is 12. The variable parts are t³, t⁶, and t². The GCF of these is t² (since we take the lowest power of t). Thus, the GCF of the three terms would be 12t², which is not 12t³. Therefore, option B is incorrect.
Option C: 30t⁴
If we add 30t⁴ to the list, the terms become 36t³, 12t⁶, and 30t⁴. The coefficients are 36, 12, and 30. To find the GCF, we can list the factors of each number. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. The GCF of 36, 12, and 30 is 6. For the variable parts, we have t³, t⁶, and t⁴. The lowest power of t is 3, so the GCF is t³. Therefore, the GCF of 36t³, 12t⁶, and 30t⁴ is 6t³, which is not 12t³. Thus, option C is incorrect.
Option D: 48t⁵
If we include 48t⁵, the terms are 36t³, 12t⁶, and 48t⁵. The coefficients are 36, 12, and 48. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. The factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The GCF of 36, 12, and 48 is 12. The variable parts are t³, t⁶, and t⁵. The lowest power of t is 3, so the GCF is t³. Thus, the GCF of the three terms 36t³, 12t⁶, and 48t⁵ is 12t³. Therefore, option D is the correct answer.
#h2 Conclusion
In conclusion, the term that can be added to the list 36t³, 12t⁶ so that the greatest common factor (GCF) of the three terms is 12t³ is 48t⁵. This was determined by finding the GCF of the coefficients and the lowest power of the variable present in all terms. By systematically evaluating each option, we identified that adding 48t⁵ to the list maintains the desired GCF of 12t³. This exercise highlights the importance of understanding the concept of GCF and how to apply it in algebraic expressions. The process involves breaking down each term into its prime factors and identifying the common factors, which is a fundamental skill in algebra and necessary for simplifying expressions and solving equations.
