Understanding the Current Terms

Oct 13, 2025 by windailys.com 62 views

Introduction

In mathematics, the greatest common factor (GCF) of a set of numbers is the largest positive integer that divides each of the numbers without leaving a remainder. When dealing with algebraic expressions, the GCF is found by identifying the common factors in the expressions and multiplying them together. In this article, we will explore how to find the GCF of three algebraic expressions and determine which term can be added to the list so that the GCF is $$12h^3$$.

Understanding the Current Terms

The current terms are $$36h^3$$ and $$12h^6$$. To find the GCF of these two expressions, we need to identify the common factors. Both expressions have a common factor of $$12h^3$$, which is the largest factor that divides both expressions without leaving a remainder. Therefore, the GCF of the current terms is $$12h^3$$.

Analyzing the Options

We are given four options to choose from: A. $$6h^3$$, B. $$12h^2$$, C. $$30h^4$$, and D. (no option provided). To determine which term can be added to the list so that the GCF is $$12h^3$$, we need to analyze each option and determine if it can be added to the list without changing the GCF.

Option A: $$6h^3$$

If we add $$6h^3$$ to the list, the new list would be $$36h^3, 12h^6, 6h^3$$. To find the GCF of these three expressions, we need to identify the common factors. The expression $$6h^3$$ has a common factor of $$6h^3$$, but the other two expressions have a common factor of $$12h^3$$. Therefore, the GCF of the new list would be $$6h^3$$, not $$12h^3$$.

Option B: $$12h^2$$

If we add $$12h^2$$ to the list, the new list would be $$36h^3, 12h^6, 12h^2$$. To find the GCF of these three expressions, we need to identify the common factors. The expression $$12h^2$$ has a common factor of $$12$$, but the other two expressions have a common factor of $$12h^3$$. Therefore, the GCF of the new list would be $$12$$, not $$12h^3$$.

Option C: $$30h^4$$

If we add $$30h^4$$ to the list, the new list would be $$36h^3, 12h^6, 30h^4$$. To find the GCF of these three expressions, we need to identify the common factors. The expression $$30h^4$$ has a common factor of $$30$$, but the other two expressions have a common factor of $$12h^3$$. Therefore, the GCF of the new list would be $$6$$, not $$12h^3$$.

Conclusion

Based on the analysis of the options, we can conclude that none of the options A, B, or C can be added to the list so that the GCF is $$12h^3$$. However, we are given a fourth option, which is not provided. To determine which term can be added to the list so that the GCF is $$12h^3$$, we need to analyze the fourth option and determine if it can be added to the list without changing the GCF.

Finding the Correct Option

To find the correct option, we need to analyze the factors of the current terms and the options. The current terms are $$36h^3$$ and $$12h^6$$, and their GCF is $$12h^3$$. To add a term to the list so that the GCF is still $$12h^3$$, the new term must have a common factor of $$12h^3$$ with the current terms.

The Correct Option

The correct option is not provided, but we can determine the correct option by analyzing the factors of the current terms and the options. The correct option must have a common factor of $$12h^3$$ with the current terms. Therefore, the correct option is a term that has a factor of $$12h^3$$.

The Final Answer

The final answer is not provided, but we can determine the correct option by analyzing the factors of the current terms and the options. The correct option is a term that has a factor of $$12h^3$$.

Final Thoughts

In conclusion, the greatest common factor of the three terms is $$12h^3$$. To add a term to the list so that the GCF is still $$12h^3$$, the new term must have a common factor of $$12h^3$$ with the current terms. Therefore, the correct option is a term that has a factor of $$12h^3$$.

Introduction

In our previous article, we discussed how to find the greatest common factor (GCF) of algebraic expressions and determined which term can be added to the list so that the GCF is $$12h^3$$. In this article, we will answer some frequently asked questions (FAQs) about the GCF of algebraic expressions.

Q&A

Q1: What is the greatest common factor (GCF) of algebraic expressions?

A1: The GCF of algebraic expressions is the largest positive integer that divides each of the expressions without leaving a remainder.

Q2: How do I find the GCF of two algebraic expressions?

A2: To find the GCF of two algebraic expressions, you need to identify the common factors in the expressions and multiply them together.

Q3: What is the difference between the GCF and the least common multiple (LCM)?

A3: The GCF is the largest positive integer that divides each of the expressions without leaving a remainder, while the LCM is the smallest positive integer that is a multiple of each of the expressions.

Q4: Can the GCF of algebraic expressions be a variable?

A4: Yes, the GCF of algebraic expressions can be a variable. For example, if the expressions are $$2x^2$$ and $$4x^2$$, the GCF is $$2x^2$$.

Q5: How do I determine which term can be added to the list so that the GCF is $$12h^3$$?

A5: To determine which term can be added to the list so that the GCF is $$12h^3$$, you need to analyze the factors of the current terms and the options. The new term must have a common factor of $$12h^3$$ with the current terms.

Q6: Can the GCF of algebraic expressions be a constant?

A6: Yes, the GCF of algebraic expressions can be a constant. For example, if the expressions are $$6x^2$$ and $$12x^2$$, the GCF is $$6$$.

Q7: How do I find the GCF of three or more algebraic expressions?

A7: To find the GCF of three or more algebraic expressions, you need to identify the common factors in the expressions and multiply them together.

Q8: Can the GCF of algebraic expressions be a polynomial?

A8: Yes, the GCF of algebraic expressions can be a polynomial. For example, if the expressions are $$x^2 + 2x$$ and $$x^2 + 4x$$, the GCF is $$x$$.

Conclusion

In conclusion, the GCF of algebraic expressions is an important concept in mathematics that can be used to simplify expressions and solve problems. By understanding how to find the GCF of algebraic expressions, you can apply this concept to a wide range of mathematical problems.

Final Thoughts

We hope this Q&A article has been helpful in answering your questions about the GCF of algebraic expressions. If you have any further questions or need additional clarification, please don't hesitate to ask.