Factor Using The Greatest Common Factor (GCF): 16 X Y 2 + 28 X Y + 8 Y 16xy^2 + 28xy + 8y 16 X Y 2 + 28 X Y + 8 Y

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Introduction

In algebra, factoring is a process of expressing a polynomial as a product of simpler polynomials. The greatest common factor (GCF) is a key concept in factoring polynomials. It is the largest expression that divides each term of the polynomial without leaving a remainder. In this article, we will learn how to factor the polynomial $16xy^2 + 28xy + 8y$ using the GCF.

Understanding the Greatest Common Factor (GCF)

The GCF of a set of numbers or expressions is the largest expression that divides each number or expression without leaving a remainder. To find the GCF, we need to identify the common factors of the numbers or expressions. In the case of the polynomial $16xy^2 + 28xy + 8y$, we need to find the common factors of the coefficients and the variables.

Finding the GCF of the Coefficients

The coefficients of the polynomial are 16, 28, and 8. To find the GCF of these coefficients, we need to identify the largest expression that divides each coefficient without leaving a remainder. The GCF of 16, 28, and 8 is 4.

Finding the GCF of the Variables

The variables of the polynomial are x and y. To find the GCF of these variables, we need to identify the largest expression that divides each variable without leaving a remainder. The GCF of x and y is 1.

Factoring the Polynomial

Now that we have found the GCF of the coefficients and the variables, we can factor the polynomial. The GCF of the coefficients is 4, and the GCF of the variables is 1. Therefore, the GCF of the polynomial is 4y.

To factor the polynomial, we need to divide each term by the GCF. The first term is $16xy^2$, which can be divided by 4y to get $4xy$. The second term is $28xy$, which can be divided by 4y to get $7x$. The third term is $8y$, which can be divided by 4y to get 2.

Therefore, the factored form of the polynomial is $4y(4xy + 7x + 2)$.

Conclusion

In this article, we learned how to factor the polynomial $16xy^2 + 28xy + 8y$ using the GCF. We found the GCF of the coefficients and the variables, and then used it to factor the polynomial. The factored form of the polynomial is $4y(4xy + 7x + 2)$. This is a key concept in algebra, and it is used to simplify complex polynomials.

Example Problems

Here are some example problems that you can try to practice factoring polynomials using the GCF:

• Factor the polynomial $24x^2y + 36xy + 12y$
• Factor the polynomial $18x^2y + 24xy + 6y$
• Factor the polynomial $20x^2y + 30xy + 10y$

Step-by-Step Solutions

Here are the step-by-step solutions to the example problems:

Factor the polynomial $24x^2y + 36xy + 12y$

1. Find the GCF of the coefficients: The GCF of 24, 36, and 12 is 12.
2. Find the GCF of the variables: The GCF of x and y is 1.
3. Factor the polynomial: Divide each term by the GCF. The first term is $24x^2y$, which can be divided by 12 to get $2x^2y$. The second term is $36xy$, which can be divided by 12 to get $3xy$. The third term is $12y$, which can be divided by 12 to get $y$.
4. Write the factored form of the polynomial: $12(2x^2y + 3xy + y)$

Factor the polynomial $18x^2y + 24xy + 6y$

1. Find the GCF of the coefficients: The GCF of 18, 24, and 6 is 6.
2. Find the GCF of the variables: The GCF of x and y is 1.
3. Factor the polynomial: Divide each term by the GCF. The first term is $18x^2y$, which can be divided by 6 to get $3x^2y$. The second term is $24xy$, which can be divided by 6 to get $4xy$. The third term is $6y$, which can be divided by 6 to get $y$.
4. Write the factored form of the polynomial: $6(3x^2y + 4xy + y)$

Factor the polynomial $20x^2y + 30xy + 10y$

1. Find the GCF of the coefficients: The GCF of 20, 30, and 10 is 10.
2. Find the GCF of the variables: The GCF of x and y is 1.
3. Factor the polynomial: Divide each term by the GCF. The first term is $20x^2y$, which can be divided by 10 to get $2x^2y$. The second term is $30xy$, which can be divided by 10 to get $3xy$. The third term is $10y$, which can be divided by 10 to get $y$.
4. Write the factored form of the polynomial: $10(2x^2y + 3xy + y)$
   Q&A: Factoring Polynomials using the Greatest Common Factor (GCF) ================================================================

Frequently Asked Questions

In this article, we will answer some frequently asked questions about factoring polynomials using the greatest common factor (GCF).

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

A: The greatest common factor (GCF) is the largest expression that divides each term of a polynomial without leaving a remainder.

Q: How do I find the GCF of a set of numbers or expressions?

A: To find the GCF of a set of numbers or expressions, you need to identify the common factors of the numbers or expressions. You can use the following steps:

1. List the numbers or expressions.
2. Identify the common factors of the numbers or expressions.
3. Find the largest common factor.

Q: How do I factor a polynomial using the GCF?

A: To factor a polynomial using the GCF, you need to follow these steps:

1. Find the GCF of the coefficients of the polynomial.
2. Find the GCF of the variables of the polynomial.
3. Divide each term of the polynomial by the GCF.
4. Write the factored form of the polynomial.

Q: What is the difference between factoring and simplifying a polynomial?

A: Factoring a polynomial involves expressing it as a product of simpler polynomials, while simplifying a polynomial involves combining like terms.

Q: Can I factor a polynomial that has no common factors?

A: No, you cannot factor a polynomial that has no common factors. In this case, the polynomial is already in its simplest form.

Q: Can I factor a polynomial that has a negative coefficient?

A: Yes, you can factor a polynomial that has a negative coefficient. You can factor the polynomial as usual, and then multiply the result by -1.

Q: Can I factor a polynomial that has a variable with a negative exponent?

A: Yes, you can factor a polynomial that has a variable with a negative exponent. You can factor the polynomial as usual, and then take the reciprocal of the result.

Q: What are some common mistakes to avoid when factoring polynomials?

A: Some common mistakes to avoid when factoring polynomials include:

• Not finding the GCF of the coefficients and variables.
• Not dividing each term of the polynomial by the GCF.
• Not writing the factored form of the polynomial correctly.

Q: How can I practice factoring polynomials?

A: You can practice factoring polynomials by:

• Working on example problems.
• Using online resources and practice exercises.
• Asking a teacher or tutor for help.

Q: What are some real-world applications of factoring polynomials?

A: Factoring polynomials has many real-world applications, including:

• Algebraic geometry.
• Number theory.
• Cryptography.

Conclusion

In this article, we have answered some frequently asked questions about factoring polynomials using the greatest common factor (GCF). We have also provided some tips and resources for practicing factoring polynomials. By following these tips and resources, you can become proficient in factoring polynomials and apply this skill to real-world problems.

Example Problems

Here are some example problems that you can try to practice factoring polynomials:

• Factor the polynomial $24x^2y + 36xy + 12y$
• Factor the polynomial $18x^2y + 24xy + 6y$
• Factor the polynomial $20x^2y + 30xy + 10y$

Step-by-Step Solutions

Here are the step-by-step solutions to the example problems:

Factor the polynomial $24x^2y + 36xy + 12y$

1. Find the GCF of the coefficients: The GCF of 24, 36, and 12 is 12.
2. Find the GCF of the variables: The GCF of x and y is 1.
3. Factor the polynomial: Divide each term by the GCF. The first term is $24x^2y$, which can be divided by 12 to get $2x^2y$. The second term is $36xy$, which can be divided by 12 to get $3xy$. The third term is $12y$, which can be divided by 12 to get $y$.
4. Write the factored form of the polynomial: $12(2x^2y + 3xy + y)$

Factor the polynomial $18x^2y + 24xy + 6y$

1. Find the GCF of the coefficients: The GCF of 18, 24, and 6 is 6.
2. Find the GCF of the variables: The GCF of x and y is 1.
3. Factor the polynomial: Divide each term by the GCF. The first term is $18x^2y$, which can be divided by 6 to get $3x^2y$. The second term is $24xy$, which can be divided by 6 to get $4xy$. The third term is $6y$, which can be divided by 6 to get $y$.
4. Write the factored form of the polynomial: $6(3x^2y + 4xy + y)$

Factor the polynomial $20x^2y + 30xy + 10y$

1. Find the GCF of the coefficients: The GCF of 20, 30, and 10 is 10.
2. Find the GCF of the variables: The GCF of x and y is 1.
3. Factor the polynomial: Divide each term by the GCF. The first term is $20x^2y$, which can be divided by 10 to get $2x^2y$. The second term is $30xy$, which can be divided by 10 to get $3xy$. The third term is $10y$, which can be divided by 10 to get $y$.
4. Write the factored form of the polynomial: $10(2x^2y + 3xy + y)$