Factoring Out the GCF: Steps and Examples

Factoring out the GCF means finding the greatest common factor of every term in a polynomial and writing it in front of parentheses. It is the distributive property run backward. For example, $6x^2 + 9x = 3x(2x + 3)$ , because $3x$ divides both terms. Always look for a GCF first when you factor.

Before this lesson: You should know how to find the greatest common factor of numbers, the distributive property, and the exponent rules.

Extra practice: The Algebra 1 Unit 7 Practice Pack (Part 1) has printable problems, answer keys and a cheat sheet for this unit.

What Factoring Means

To factor means to write something as a product. You already factor numbers: $12 = 4 \cdot 3$ . Now you will factor polynomials.

When you distribute, you multiply: $3x(2x + 3) = 6x^2 + 9x$ . When you factor, you go the other way. You start with $6x^2 + 9x$ and find the $3x(2x + 3)$ it came from.

A rectangle split into two parts with areas 6x² and 9x. The shared height is 3x, the GCF. The widths of the parts are 2x and 3. Beside it: multiplying 3x(2x + 3) gives 6x² + 9x, and factoring 6x² + 9x gives 3x(2x + 3). — The two areas share a side of 3x. Factoring finds that side and the other side, 2x + 3.

Think of the polynomial as the area of a rectangle. The GCF is one side. The part in parentheses is the other side.

How to Factor Out the GCF

The greatest common factor (GCF) of a polynomial is the largest monomial that divides every term. It has two parts.

Part of the GCF	How to find it	Example: $12x^3$ and $18x^2$
Number	The GCF of the coefficients	$6$
Each variable	The smallest exponent that appears in every term	$x^2$
Whole GCF	Multiply the two parts	$6x^2$

So $12x^3 + 18x^2 = 6x^2(2x + 3)$ . To find each term inside, divide by the GCF: $\frac{12x^3}{6x^2} = 2x$ and $\frac{18x^2}{6x^2} = 3$ .

Steps

Find the GCF of the coefficients.
For each variable that is in every term, take the smallest exponent.
Write the GCF in front of a set of parentheses.
Divide each term by the GCF. Write the answers inside the parentheses.
Check by distributing. You should get the original polynomial.

Worked Examples

Example 1: A number GCF

Factor $8x + 12$ .

Step 1. Find the greatest common factor of the coefficients. The factors of 8 are 1, 2, 4, 8, and the factors of 12 are 1, 2, 3, 4, 6, 12. The largest shared factor is 4.

Step 2. Check the variables. Only the first term has an $x$ , so $x$ is not part of the GCF. The GCF is 4.

8x + 12 = 4 \cdot 2x + 4 \cdot 3

Step 3. Divide each term by 4: $\frac{8x}{4} = 2x$ and $\frac{12}{4} = 3$ . Write each term as 4 times its quotient.

= 4(2x + 3)

Step 4. Pull out the 4. This is the distributive property in reverse.

Answer: $4(2x + 3)$

Check: Distribute: $4 \cdot 2x + 4 \cdot 3 = 8x + 12$ .

Example 2: A variable in the GCF

Factor $10x^3 - 15x^2$ .

Step 1. Find the GCF of the coefficients. The GCF of 10 and 15 is 5.

Step 2. Find the smallest power of $x$ . The powers are $x^3$ and $x^2$ , so use $x^2$ . The GCF is $5x^2$ .

10x^3 - 15x^2 = 5x^2 \cdot 2x - 5x^2 \cdot 3

Step 3. Divide each term by $5x^2$ . Subtract the exponents (see exponent rules): $\frac{10x^3}{5x^2} = 2x$ and $\frac{15x^2}{5x^2} = 3$ .

= 5x^2(2x - 3)

Step 4. Pull out $5x^2$ .

Answer: $5x^2(2x - 3)$

Check: Distribute: $5x^2 \cdot 2x - 5x^2 \cdot 3 = 10x^3 - 15x^2$ .

Example 3: Two variables and three terms

Factor $12a^3b^2 + 18a^2b^3 - 6a^2b$ .

Step 1. Find the GCF of the coefficients. The GCF of 12, 18 and 6 is 6.

Step 2. Find the smallest power of each variable. For $a$ : $a^3$ , $a^2$ , $a^2$ , so use $a^2$ . For $b$ : $b^2$ , $b^3$ , $b$ , so use $b$ . The GCF is $6a^2b$ .

= 6a^2b \cdot 2ab + 6a^2b \cdot 3b^2 - 6a^2b \cdot 1

Step 3. Divide each term by $6a^2b$ : the quotients are $2ab$ , $3b^2$ and 1.

= 6a^2b(2ab + 3b^2 - 1)

Step 4. Pull out $6a^2b$ . The last term divided by the GCF is 1. Don’t leave it out. The parentheses must still have three terms.

Answer: $6a^2b(2ab + 3b^2 - 1)$

Example 4: A negative leading coefficient

Factor $-4x^2 + 20x$ .

Step 1. The GCF of 4 and 20 is 4, and both terms have at least one x. So the GCF is 4x.

Step 2. The first term is negative, so factor out a negative GCF, $-4x$ . Then the first term inside will be positive.

-4x^2 + 20x = -4x \cdot x + (-4x)(-5)

Step 3. Divide each term by $-4x$ : $\frac{-4x^2}{-4x} = x$ and $\frac{20x}{-4x} = -5$ . Dividing by a negative changes the sign (see multiplying and dividing integers).

= -4x(x - 5)

Step 4. Pull out $-4x$ .

Answer: $-4x(x - 5)$

Check: Distribute: $-4x \cdot x + (-4x)(-5) = -4x^2 + 20x$ .

Example 5: Word problem

A rectangular patio has an area of $12x^2 + 8x$ square feet. Its width is $4x$ feet. Find its length. Then find the length and area when $x = 3$ .

Step 1. Area is width times length. So factor the width, $4x$ , out of the area. It is also the GCF.

12x^2 + 8x = 4x \cdot 3x + 4x \cdot 2

Step 2. Divide each term by $4x$ : $\frac{12x^2}{4x} = 3x$ and $\frac{8x}{4x} = 2$ .

= 4x(3x + 2)

Step 3. Pull out $4x$ .

4(3) = 12

Step 4. The other factor is the length: $3x + 2$ feet. Substitute 3 for x in the width.

3(3) + 2 = 11

Step 5. Substitute 3 for x in the length.

12 \cdot 11 = 132

Step 6. Multiply to find the area.

Answer: The length is $3x + 2$ feet. When $x = 3$ , the patio is 12 feet by 11 feet, and its area is 132 square feet.

Check: Put $x = 3$ into the area: $12(3)^2 + 8(3) = 12(9) + 8(3) = 108 + 24 = 132$ .

Common Mistakes

Mistake	Example of the error	Fix
Leaving out the 1	$6x^2 + 3x = 3x(2x)$	$\frac{3x}{3x} = 1$ , so $3x(2x + 1)$ .
Not using the greatest factor	$12x + 18 = 2(6x + 9)$	6 and 9 still share 3. Use 6: $6(2x + 3)$ .
Using the largest exponent	$x^4 + x^2 = x^4(1 + x^2)$	Use the smallest exponent: $x^2(x^2 + 1)$ .
Sign errors with a negative GCF	$-3x - 6 = -3(x - 6)$	$\frac{-6}{-3} = 2$ , so $-3(x + 2)$ .
Dividing only some terms	$5x^2 + 10x - 15 = 5(x^2 + 10x - 15)$	Divide every term: $5(x^2 + 2x - 3)$ .

More Ways the GCF Shows Up

When the GCF is a binomial

Sometimes every term shares a whole binomial. Treat it like one big factor.

3x(x + 2) + 5(x + 2) = (x + 2)(3x + 5)

Both terms have the factor $(x + 2)$ . Pull it out, and what is left is $3x + 5$ . You will use this idea again in factoring by grouping.

Factor out the GCF first

Taking out the GCF first makes the rest easier to factor. For example, $2x^2 + 10x + 12 = 2(x^2 + 5x + 6)$ . The trinomial inside has smaller numbers, so it is easier to factor in the next lesson, Factoring Trinomials.

When the GCF is 1

Some polynomials, like $3x + 5$ or $x^2 + 7$ , have no common factor except 1. You can’t pull out a GCF. Some of them may still factor in other ways.

Factoring also undoes the special products of binomials and other polynomial products, as you will see in this unit.

Practice Problems

Factor $7x + 21$ .
Factor $9y^2 - 6y$ .
Factor $4x^3 + 8x^2 - 12x$ .
Factor $15a^2b - 25ab^2$ .
Factor $2x^4 + 6x^3 + 2x^2$ .
Factor $-6x^2 - 18x$ .
Factor $14m^3n^2 + 21m^2n^2 - 7mn^2$ .
Factor $x(x - 4) + 9(x - 4)$ .
Find the GCF of $24x^3$ , $36x^2$ and $60x$ .
A rectangle has an area of $18x^2 + 27x$ square meters and a width of $9x$ meters. Find its length. Then find the area when $x = 2$ .

Show answers

Find the GCF: 7 divides both 7 and 21, and only one term has x, so the GCF is 7

7 \cdot x + 7 \cdot 3

Divide each term by 7.

7(x + 3)

Pull out the GCF.

Find the GCF: the GCF of 9 and 6 is 3, and the smallest power of y is $y$ . The GCF is $3y$

3y \cdot 3y - 3y \cdot 2

Divide each term by 3y.

3y(3y - 2)

Pull out the GCF.

Check by distributing: $3y(3y - 2) = 9y^2 - 6y$

Find the GCF: the GCF of 4, 8 and 12 is 4, and the smallest power of x is $x$ . The GCF is $4x$

4x \cdot x^2 + 4x \cdot 2x - 4x \cdot 3

Divide each term by 4x.

4x(x^2 + 2x - 3)

Pull out the GCF.

Find the GCF: the GCF of 15 and 25 is 5, the smallest power of a is $a$ , and the smallest power of b is $b$ . The GCF is $5ab$

5ab \cdot 3a - 5ab \cdot 5b

Divide each term by 5ab.

5ab(3a - 5b)

Pull out the GCF.

Find the GCF: the GCF of 2, 6 and 2 is 2, and the smallest power of x is $x^2$ . The GCF is $2x^2$

2x^2 \cdot x^2 + 2x^2 \cdot 3x + 2x^2 \cdot 1

Divide each term by $2x^2$ . The last quotient is 1.

2x^2(x^2 + 3x + 1)

Pull out the GCF.

Find the GCF: the GCF of 6 and 18 is 6, and both terms have x. The first term is negative, so use $-6x$

-6x \cdot x + (-6x)(3)

Divide each term by $-6x$ . Both signs change.

-6x(x + 3)

Pull out the GCF.

Find the GCF: the GCF of 14, 21 and 7 is 7, the smallest power of m is $m$ , and every term has $n^2$ . The GCF is $7mn^2$

7mn^2 \cdot 2m^2 + 7mn^2 \cdot 3m - 7mn^2 \cdot 1

Divide each term by the GCF. The last quotient is 1.

7mn^2(2m^2 + 3m - 1)

Pull out the GCF.

Both terms have the same binomial factor, $x - 4$ . That binomial is the GCF

(x - 4)(x + 9)

Pull it out. What is left from the two terms is $x$ and $+9$ .

Find the GCF of 24, 36 and 60. The factors of 24 that also divide 36 and 60 are 1, 2, 3, 4, 6 and 12. The largest is 12
Find the smallest power of x: $x^3$ , $x^2$ , $x$ . The smallest is $x^1 = x$
The GCF is $12x$ .

10.

18x^2 + 27x = 9x \cdot 2x + 9x \cdot 3

Area is width times length, so factor out the width, $9x$ .

9x(2x + 3)

Pull out 9x.

9(2) = 18

The length is $2x + 3$ meters. Substitute 2 for x in the width.

2(2) + 3 = 7

Substitute 2 for x in the length.

18 \cdot 7 = 126

Multiply to find the area.

Check with the area expression: $18(2)^2 + 27(2) = 72 + 54 = 126$
The area is 126 square meters.

Want more practice? The Algebra 1 Unit 7 Practice Pack (Part 1) has 37 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

FAQ

What does it mean to factor out the GCF?

It means you divide every term by their greatest common factor and write the GCF in front of parentheses. For example, $5x + 10 = 5(x + 2)$ .

How do you find the GCF of a polynomial?

Find the GCF of the coefficients. Then, for each variable in every term, use the smallest exponent. Multiply these together.

Should I factor out a negative?

Usually yes, when the first term is negative. Factoring out a negative GCF leaves a positive first term, like $-2x^2 + 8x = -2x(x - 4)$ . Remember to change every sign inside.

What if there is no common factor?

Then the GCF is 1, and you can’t pull anything out. The polynomial may still factor another way, such as a trinomial like $x^2 + 5x + 6$ .

Summary

Factoring writes a polynomial as a product. It undoes distributing.
The GCF uses the GCF of the coefficients and the smallest power of each shared variable.
Divide every term by the GCF and write the results in parentheses.
If a term equals the GCF, a 1 is left in its place.
Check by distributing. Factor out the GCF first in every factoring problem.

Previous lesson: Special Products of Binomials

Next lesson: Factoring Trinomials

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