Identifying the Terms in the Expression

Oct 13, 2025 by magicalceylontours.com 81 views

Understanding the Concept of Greatest Common Factor (GCF)

The greatest common factor (GCF) is a mathematical concept that refers to the largest expression that divides each term of a polynomial expression without leaving a remainder. In other words, it is the largest factor that is common to all the terms of the expression.

Identifying the Terms in the Expression

To find the GCF of the terms in the expression $$21x - 18x + 28xy$$, we need to identify the individual terms. The terms in the expression are:

$21x$
$-18x$
$28xy$

Breaking Down the Terms

To find the GCF, we need to break down each term into its prime factors. The prime factorization of each term is:

$21x = 3 \times 7 \times x$
$-18x = -2 \times 3^2 \times x$
$28xy = 2^2 \times 7 \times x \times y$

Finding the Greatest Common Factor

Now that we have broken down each term into its prime factors, we can find the GCF by identifying the common factors among the terms. The common factors among the terms are:

Determining the Greatest Common Factor

Since the GCF must be a single expression, we need to combine the common factors. The GCF of the terms in the expression $$21x - 18x + 28xy$$ is therefore:

\boxed{3x}

Conclusion

In conclusion, the greatest common factor of the terms in the expression $$21x - 18x + 28xy$$ is $$3x$$. This is because $$3x$$ is the largest expression that divides each term of the polynomial expression without leaving a remainder.

Frequently Asked Questions

What is the greatest common factor (GCF)? The GCF is the largest expression that divides each term of a polynomial expression without leaving a remainder.
How do I find the GCF of a polynomial expression? To find the GCF, you need to break down each term into its prime factors and identify the common factors among the terms.
What is the GCF of the terms in the expression $$21x - 18x + 28xy$$? The GCF of the terms in the expression $$21x - 18x + 28xy$$ is $$3x$$.

Final Answer

The final answer is: $\boxed{3x}$

Q&A: Greatest Common Factor (GCF)

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

A1: The greatest common factor (GCF) is a mathematical concept that refers to the largest expression that divides each term of a polynomial expression without leaving a remainder.

Q2: How do I find the GCF of a polynomial expression?

A2: To find the GCF, you need to break down each term into its prime factors and identify the common factors among the terms.

Q3: What is the GCF of the terms in the expression $$21x - 18x + 28xy$$?

A3: The GCF of the terms in the expression $$21x - 18x + 28xy$$ is $$3x$$.

Q4: What is the difference between GCF and Least Common Multiple (LCM)?

A4: The GCF is the largest expression that divides each term of a polynomial expression without leaving a remainder, while the LCM is the smallest expression that is a multiple of each term of a polynomial expression.

Q5: How do I use the GCF to simplify a polynomial expression?

A5: To simplify a polynomial expression using the GCF, you need to factor out the GCF from each term of the expression.

Q6: What is the GCF of two or more polynomial expressions?

A6: The GCF of two or more polynomial expressions is the product of the GCFs of each expression.

Q7: Can the GCF be a constant?

A7: Yes, the GCF can be a constant. For example, the GCF of the terms $$2x$$ and $$4x$$ is $$2$$.

Q8: Can the GCF be a variable?

A8: Yes, the GCF can be a variable. For example, the GCF of the terms $$x$$ and $$xy$$ is $$x$$.

Q9: How do I find the GCF of a polynomial expression with multiple variables?

A9: To find the GCF of a polynomial expression with multiple variables, you need to break down each term into its prime factors and identify the common factors among the terms.

Q10: What is the importance of finding the GCF in mathematics?

A10: Finding the GCF is important in mathematics because it helps to simplify polynomial expressions, which can make it easier to solve equations and inequalities.

Final Answer

The final answer is: $\boxed{3x}$