Find The Greatest Common Factor (GCF) Of The Monomials 6 N 3 6n^3 6 N 3 And 2 2 2 .Write Your Answer As A Constant Times A Product Of Single Variables Raised To Exponents. □ \square □ Submit

What is the Greatest Common Factor (GCF)?

The greatest common factor (GCF) of two or more monomials is the product of the common factors of the highest power that divides each monomial without leaving a remainder. In other words, it is the largest monomial that divides each of the given monomials without leaving a remainder.

Understanding Monomials

A monomial is an algebraic expression that consists of a single term, which is a product of a coefficient and one or more variables raised to non-negative integer exponents. For example, $3x^2$, $2y^3$, and $5z$ are all monomials.

Finding the GCF of Monomials

To find the GCF of two monomials, we need to identify the common factors of the highest power that divides each monomial without leaving a remainder. Let's consider the monomials $6n^3$ and $2$.

Step 1: Identify the Common Factors

The monomial $6n^3$ has a coefficient of $6$ and a variable of $n$ raised to the power of $3$. The monomial $2$ has a coefficient of $2$ and no variable.

Step 2: Identify the Common Variable

Since the monomial $2$ has no variable, we can conclude that there is no common variable between the two monomials.

Step 3: Identify the Common Coefficient

The monomial $6n^3$ has a coefficient of $6$, while the monomial $2$ has a coefficient of $2$. Since $2$ is a factor of $6$, we can conclude that the common coefficient is $2$.

Step 4: Write the GCF as a Product of Single Variables Raised to Exponents

Since there is no common variable, the GCF is simply the common coefficient, which is $2$.

Conclusion

The greatest common factor (GCF) of the monomials $6n^3$ and $2$ is $2$.

Example

Find the GCF of the monomials $4x^2y$ and $2xy$.

Step 1: Identify the Common Factors

The monomial $4x^2y$ has a coefficient of $4$ and variables of $x$ raised to the power of $2$ and $y$. The monomial $2xy$ has a coefficient of $2$ and variables of $x$ and $y$.

Step 2: Identify the Common Variable

Since both monomials have the variable $y$, we can conclude that the common variable is $y$.

Step 3: Identify the Common Coefficient

The monomial $4x^2y$ has a coefficient of $4$, while the monomial $2xy$ has a coefficient of $2$. Since $2$ is a factor of $4$, we can conclude that the common coefficient is $2$.

Step 4: Write the GCF as a Product of Single Variables Raised to Exponents

Since the common variable is $y$ and the common coefficient is $2$, the GCF is $2y$.

Conclusion

The greatest common factor (GCF) of the monomials $4x^2y$ and $2xy$ is $2y$.

Tips and Tricks

• When finding the GCF of monomials, always identify the common factors, variables, and coefficients.
• Use the distributive property to factor out common factors.
• Simplify the GCF by combining like terms.

Practice Problems

Find the GCF of the following monomials:

1. $3x^2y$ and $2xy$
2. $4x^3y^2$ and $2x^2y$
3. $6x^2y^3$ and $3xy^2$

Answer Key

1. $2xy$
2. $2x^2y$
3. $3xy^2$
   Greatest Common Factor (GCF) of Monomials: Q&A =====================================================

Frequently Asked Questions

Q: What is the greatest common factor (GCF) of two monomials?

A: The greatest common factor (GCF) of two monomials is the product of the common factors of the highest power that divides each monomial without leaving a remainder.

Q: How do I find the GCF of two monomials?

A: To find the GCF of two monomials, you need to identify the common factors, variables, and coefficients. Then, use the distributive property to factor out common factors and simplify the GCF by combining like terms.

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

A: The greatest common factor (GCF) is the product of the common factors of the highest power that divides each monomial without leaving a remainder, while the least common multiple (LCM) is the product of the common factors of the highest power that is a multiple of each monomial.

Q: Can the GCF of two monomials be a variable?

A: No, the GCF of two monomials cannot be a variable. The GCF is always a product of coefficients and variables raised to non-negative integer exponents.

Q: Can the GCF of two monomials be a constant?

A: Yes, the GCF of two monomials can be a constant. For example, the GCF of $6x^2$ and $2x^2$ is $2x^2$, which is a constant times a variable raised to an exponent.

Q: How do I find the GCF of more than two monomials?

A: To find the GCF of more than two monomials, you need to find the GCF of the first two monomials, and then find the GCF of the result and the third monomial, and so on.

Q: Can the GCF of two monomials be zero?

A: No, the GCF of two monomials cannot be zero. The GCF is always a non-zero product of coefficients and variables raised to non-negative integer exponents.

Q: What is the importance of finding the GCF of monomials?

A: Finding the GCF of monomials is important in algebra because it helps to simplify expressions and solve equations. It is also used in calculus to find the derivative and integral of functions.

Common Mistakes to Avoid

• Not identifying the common factors, variables, and coefficients.
• Not using the distributive property to factor out common factors.
• Not simplifying the GCF by combining like terms.
• Assuming that the GCF of two monomials is a variable or a constant without checking.

Tips and Tricks

• Use the distributive property to factor out common factors.
• Simplify the GCF by combining like terms.
• Check if the GCF is a variable or a constant before simplifying.
• Use the GCF to simplify expressions and solve equations.

Practice Problems

Find the GCF of the following monomials:

1. $3x^2y$ and $2xy$
2. $4x^3y^2$ and $2x^2y$
3. $6x^2y^3$ and $3xy^2$

Answer Key

1. $2xy$
2. $2x^2y$
3. $3xy^2$

Real-World Applications

• Finding the GCF of monomials is used in algebra to simplify expressions and solve equations.
• It is also used in calculus to find the derivative and integral of functions.
• In engineering, the GCF of monomials is used to design and optimize systems.
• In economics, the GCF of monomials is used to model and analyze economic systems.

Conclusion

The greatest common factor (GCF) of monomials is an important concept in algebra that helps to simplify expressions and solve equations. By understanding the GCF, you can apply it to real-world problems and make informed decisions. Remember to identify the common factors, variables, and coefficients, and use the distributive property to factor out common factors. Simplify the GCF by combining like terms, and check if it is a variable or a constant before simplifying.