# Factoring Polynomials Using the Greatest Common Factor (GCF)

### Definition of Factors

Factoring a polynomial involves finding the quantities that, when multiplied together, produce the original polynomial. Factors are numbers or expressions that divide evenly into a given number or expression.

### Key Facts

1. Definition of Factors:
• When factoring a polynomial, you are trying to find the quantities that you multiply together to create the polynomial.
• Factors are the numbers or expressions that divide evenly into a given number or expression.
2. Greatest Common Factor (GCF):
• The GCF for a polynomial is the largest monomial that is a factor of each term in the polynomial.
• The GCF must be a factor of every term in the polynomial.
• Finding the GCF involves identifying the highest power of each variable that appears in all the terms.
3. Process of Factoring Using GCF:
• Find the GCF of all the terms in the polynomial.
• Express each term as a product of the GCF and another factor.
• Use the distributive property to factor out the GCF.
4. Examples:
• Example 1: Factoring Using the GCF
• Polynomial: 6x^2 + 9x
• GCF: 3x
• Factored form: 3x(2x + 3)
• Example 2: Factoring Polynomials
• Polynomial: 12x^3y^2 + 8xy^3
• GCF: 4xy^2
• Factored form: 4xy^2(3x^2 + 2y)
• Example 3: Factoring Polynomials
• Polynomial: 15a^2b – 10ab^2
• GCF: 5ab
• Factored form: 5ab(3a – 2b)

### Greatest Common Factor (GCF)

The GCF of a polynomial is the largest monomial that is a factor of every term in the polynomial. The GCF must be a factor of every term in the polynomial. To find the GCF, identify the highest power of each variable that appears in all the terms.

### Process of Factoring Using GCF

1. Determine the GCF of all the terms in the polynomial.
2. Express each term as a product of the GCF and another factor.
3. Use the distributive property to factor out the GCF.

## Example 1: Factoring Using the GCF

• Polynomial: 6x^2 + 9x
• GCF: 3x
• Factored form: 3x(2x + 3)

## Example 2: Factoring Polynomials

• Polynomial: 12x^3y^2 + 8xy^3
• GCF: 4xy^2
• Factored form: 4xy^2(3x^2 + 2y)

## Example 3: Factoring Polynomials

• Polynomial: 15a^2b – 10ab^2
• GCF: 5ab
• Factored form: 5ab(3a – 2b)

## FAQs

### What is factoring by GCF?

Factoring by GCF involves finding the greatest common factor (GCF) of all the terms in a polynomial and then expressing each term as a product of the GCF and another factor. The GCF is then factored out using the distributive property.

### How do I find the GCF of a polynomial?

To find the GCF of a polynomial, identify the highest power of each variable that appears in all the terms. The GCF is the product of these highest powers.

### When can I use factoring by GCF?

Factoring by GCF can be used when every term in the polynomial has a common factor.

### What are the steps for factoring by GCF?

1. Find the GCF of all the terms in the polynomial.
2. Express each term as a product of the GCF and another factor.
3. Use the distributive property to factor out the GCF.

### Can I factor any polynomial using GCF?

No, factoring by GCF is only possible when every term in the polynomial has a common factor.

### What are some examples of factoring by GCF?

• 6x^2 + 9x = 3x(2x + 3)
• 12x^3y^2 + 8xy^3 = 4xy^2(3x^2 + 2y)
• 15a^2b – 10ab^2 = 5ab(3a – 2b)

### What are the benefits of factoring by GCF?

Factoring by GCF can simplify polynomials, make them easier to solve, and reveal important properties of the polynomial.

### What are some common mistakes to avoid when factoring by GCF?

• Not finding the GCF correctly
• Not expressing each term as a product of the GCF and another factor
• Not using the distributive property correctly