Math resources Algebra Factoring

Factoring out the GCF

# Factoring out the GCF

Here you will learn about factoring out the GCF, including what the GCF is and how to factor out the GCF with different polynomials.

Students will first learn about factoring out the GCF as a part of expressions and equations in grades 6 and 7. Students will expand on their knowledge pre-algebra and algebra 1.

## What is factoring out the GCF?

Factoring out the GCF means to take out a common factor from algebraic expressions in order to put it into parenthesis.

Factoring is writing the algebraic expression as a product of its factors.

The greatest common factor, or GCF, is the largest number that is a factor of 2 or more terms.

For example, the GCF of 16x+12 is 4 so the factored form is 4(4x+3). You can check this by multiplying out the parentheses using the distributive property: 4\times{4x}+4\times{3}=16x+12.

Another common factor of 16x+12 is 2 but this is not the greatest common factor.

The GCF can contain both numbers and variables. Example, the GCF of 12x^{2} and 8xy is 4x.

### Factoring polynomials

A polynomial is an algebraic expression with 2 or more terms, or a sum of one or more monomials. The terms do not include exponents that are negative or fractions.

A monomial is a polynomial containing one term which may be a number or variable.

For example,

3x^{2} is a monomial,

x+5 is a binomial (a polynomial containing two unlike terms)

2x^{2}+x+3y is a trinomial (a polynomial containing 3 or more unlike terms)

## Common Core State Standards

How does this relate to 6 th grade math and 7 th grade math?

• Grade 6: Expressions and equations (6.EE.A.3)
Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2+x) to produce the equivalent expression 6+3x; apply the distributive property to the expression 24x+18y to produce the equivalent expression 6(4x+3y); apply properties of operations to y+y+y to produce the equivalent expression 3y.

• Grade 7: Expressions and equations (7.EE.A.1)
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

## How to factor out the GCF

In order to factor out the GCF, you will need to:

1. Find the GCF of all terms in the expression and place it in front of parentheses.
2. Divide each term in the expression by the GCF.
3. Put the divided terms in the parentheses.
4. Express the GCF and the other factor(s) as products.

## Factoring out the GCF examples

### Example 1: factoring out GCF

Factor the algebraic expression.

3x+6

1. Find the GCF of all terms in the expression and place it in front of parentheses.

List all the factor pairs of 3 and 6 separately:

The greatest common factor in the expression is 3.

2Divide each term in the expression by the GCF.

Since 3 is the GCF,

3x\div{3}=x

6\div{3}=2

3Put the divided terms in the parentheses.

3(x+2)

4Express the GCF and the other factor(s) as products.

3x+6=3(x+2)

### Example 2: variable in just one term

Factor the algebraic expression 14-7y.

Find the GCF of all terms in the expression and place it in front of parentheses.

Divide each term in the expression by the GCF.

Put the divided terms in the parentheses.

Express the GCF and the other factor(s) as products.

### Example 3: variable in two terms

Factor the binomial 8x^{2}+12x .

Find the GCF of all terms in the expression and place it in front of parentheses.

Divide each term in the expression by the GCF.

Put the divided terms in the parentheses.

Express the GCF and the other factor(s) as products.

### Example 4: variables in two terms

Factor the binomial 15y^{2}-10xy.

Find the GCF of all terms in the expression and place it in front of parentheses.

Divide each term in the expression by the GCF.

Put the divided terms in the parentheses.

Express the GCF and the other factor(s) as products.

### Example 5: factoring trinomials

Factor the trinomial 6x+2y-12.

Find the GCF of all terms in the expression and place it in front of parentheses.

Divide each term in the expression by the GCF.

Put the divided terms in the parentheses.

Express the GCF and the other factor(s) as products.

### Example 6: variables in three out of three terms

Factor the algebraic expression: 12xy-4x^{3}y+8xy^{2}

Find the GCF of all terms in the expression and place it in front of parentheses.

Divide each term in the expression by the GCF.

Put the divided terms in the parentheses.

Express the GCF and the other factor(s) as products.

### Teaching tips for factoring out the GCF

• When worksheets with practice problems need to be used, consider having students work in pairs or small groups to solve the problems. This allows students to explain their reasoning when finding the right answer and work together to solve each problem.

• Provide students with step by step guides and example problems to refer back to.

• Incorporate games when teaching this topic.

### Easy mistakes to make

• Making sign errors when factoring out the GCF
It is easy to forget to include the sign when factoring out the GCF. In order to double check your factoring, you can use the distributive property to ensure you have used the correct signs and factored correctly.

• Not factoring the GCF out completely
When factoring, make sure to find the true GCF of all terms.
For example, 12x^{2}-6x=2\left(6x^2-3x\right). Here the expression has been factored, however it is not fully factored because 2 is not the greatest common factor. 6x is the GCF, so the correct final answer is: 12x^{2}-6x=6x(2x-1).

### Practice factoring out the GCF questions

1. Factor the algebraic expression, 5x+10 .

5(x+10)

5x(x+2)

5(x+2)

x(5+10)

The greatest common factor of 5 and 10 is 5.

5x\div{5}=x

10\div{5}=2

5 x+10=5(x+2)

2. Factor the algebraic expression,

8-2y .

2(4-y)

2(4+y)

2y(4-y)

8(1-2y)

The greatest common factor of 8 and 2 is 2.

8\div{2}=4

– \, 2y\div{2}=-y

8-2y=2(4-y)

3. Factor the algebraic expression, 18x^{2}-12x.

3x(6x-4)

6\left(3x^{2}-2x\right)

3\left(6x^{2}-4x\right)

6x(3x-2)

The greatest common factor of 18x^{2} and 12x is 6x.

Divide each term in the original expression by 6x.

18x^{2}\div{6x}=3x

– \, 12x\div{6x}=-2

18x^{2}-12x=6x(3x-2)

4. Factor the algebraic expression, 20y^{2}+16xy .

4y(5y+4x)

y(20y-16x)

2y(10y-8x)

4xy(5y-4x)

The greatest common factor of 20y^{2} and 16xy is 4y.

Divide each term in the original expression by 4y.

20y^{2}\div{4y}=5y

16xy\div{4y}=4x

20y^{2}+16xy=4y(5y+4x)

5. Factor the algebraic expression,

18-6y+15x .

3xy(6-2y+5x)

3(6-2y+5x)

18(1-2y+5x)

3(6+2y+5x)

The greatest common factor of 18, \, 6y and 15x is 3.

Divide each term in the original expression by 3.

18\div{3}=6

– \, 6y\div{3}=- \, 2y

15x\div{3}=5x

18-6y+15x=3(6-2y+5x)

6. Factor the algebraic expression, 12y-9x^{2}y+6y^{2} .

12y\left(1-3x^{2}+2y\right)

3y\left(4+3x^{2}+2y\right)

3y\left(4-3x^{2}+2y\right)

3y\left(4-3 x^{2}-2y\right)

The greatest common factor of 12y, \, 9x^{2}y and 6y^{2} is 3y.

Divide each term in the original expression by 3y.

12y\div{3y}=4

– \, 9x^2y\div{3y}=- \, 3x^2

6y^{2}\div{3y}=2y

12y-9x^{2}y+6y^{2}=3y\left(4-3x^{2}+2y\right)

## Factoring out the GCF FAQs

What is the greatest common factor (GCF)?

The GCF is the largest factor that can be divided from all terms within an expression. It can include both numerical coefficients and any variable.

What’s the difference between the greatest common factor (GCF) and the least common multiple (LCM)?

The greatest common factor (GCF) and least common multiple (LCM) are often confused within the math world. The GCF is the largest positive factor that divides all the numbers without leaving a remainder. The LCM is the smallest positive multiple that is divisible by all the numbers without leaving a remainder.

How can you use prime factorization to find the GCF?

The GCF is the product of the smallest power of all the common prime factors. For example, find the GCF of 18 and 24 using prime factorization.

18=2\times{3^2}

24=2^{3}\times{3}

The common prime factors are 2 and 3.

GCF=2 \times 3=6

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