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Mathematics LibreTexts

2.7: The Greatest Common Factor and Factor by Grouping

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We begin this section with definition of a factor of a term. Because 24=212, both 2 and 12 are factors of 24. Note that 2 is also a divisor of 24, because when you divide 24 by 2 you get 12, with a remainder of zero. Similarly, 12 is also a divisor of 24, because when you divide 24 by 12 you get 2, with a remainder of zero.

Factors of a Term

Suppose m and n are integers. Then m is a factor of a term, n, if and only if there exists another integer k so that n=mk.

A factor is also known as a divisor of a term since the term can be divided evenly by the factor, nm=k.

Example 2.7.1

List the factors of 24.

Solution

First, list all possible ways that we can express 24 as a product of two positive integers:

24=124 or 24=212 or 24=38 or 24=46

Therefore, the factors of 24 are 1,2,3,4,6,8, and 24.

You Try 2.7.1

List the factors of 18.

Answer

1,2,3,6,9, and 18

Example 2.7.2

List the factors that 36 and 48 have in common.

Solution

First, list all factors of 36 and 48 separately, then determine the factors that are in common.

Factors of 36 are: 1,2,3,4,6,9,12,18,36

Factorss of 48 are: 1,2,3,4,6,8,12,16,24,48

Therefore, the common factors of 36 and 48 are 1,2,3,4,6, and 12.

You Try 2.7.2

List the factors that 40 and 60 have in common.

Answer

1,2,4,5,10, and 20

Greatest Common Factor

The greatest common factor of a and b is the largest positive number that divides evenly (with no remainder) both a and b. The greatest common factor of a and b is denoted by the symbolism GCF(a,b).

Since factors are also divisors, the abbreviation GCD(a,b) also represents the greatest common factor of a and b.

In Example 2.7.2, we listed the common factors of 36 and 48. The largest of these common factor was 12. Hence, the greatest common factor of 36 and 48 is 12, written GCF(36,48)=12.

With larger numbers, it can be more difficult to identify the greatest common factor. Prime factorization can be helpful in this situation!

Example 2.7.3

Find the greatest common factor of 360 and 756.

Solution

Prime factor 360 and 756, writing your answer in exponential form.

Diagrams of a prime factorization tree for 360 and 756
Figure 2.7.1

Thus:

360=23325756=22337

To find the greatest common factor, list each factor that appears in common and the highest power that appears in common amongst the terms.

In this case, the factors 2 and 3 appear in common, with 2 being the highest power of 2 and 2 being the highest power of 3 that appear in common. Therefore, the greatest common factor of 360 and 756 is:

GCF(360,756)=2232=49=36

Therefore, the greatest common factor is GCF(360,756)=36.

Note what happens when we write each of the given numbers in the last example as a product of the greatest common factor and a second factor:

360=3610756=3621

In each case, note how the second second factors (10 and 21) contain no additional common factors.

You Try 2.7.3

Find the greatest common factor of 120 and 450.

Answer

30

Finding the Greatest Common Factor of Monomials

Example 2.7.3 reveals the technique used to find the greatest common factor of two or more monomials.

Finding the Greatest Common Factor (GCF) of Two or More Monomials

To find the greatest common factor of two or more monomials:

  1. Find the greatest common factor of the coefficients of the given monomials. Use prime factorization if necessary.
  2. List each variable that appears in common in the given monomials.
  3. Raise each variable factor that appears in common to the highest power that appears in common among the given monomials.
Example 2.7.4

Find the greatest common factor of 6x3y3 and 9x2y5.

Solution

To find the GCF of 6x3y3 and 9x2y5, we note that:

  1. The greatest common factor of 6 and 9 is 3.
  2. The monomials 6x3y3 and 9x2y5 have the variables x and y in common.
  3. The highest power of x in common is x2. The highest power of y in common is y3.

Thus, the greatest common factor is GCF(6x3y3,9x2y5)=3x2y3. Note what happens when we write each of the given monomials as a product of the greatest common factor and a second monomial:

6x3y3=3x2y32x9x2y5=3x2y33y

Observe that the set of second monomial factors (2x and 3y) contain no additional common factors.

You Try 2.7.4

Find the greatest common factor of 16xy3 and 12x4y2.

Answer

4xy2

Example 2.7.5

Find the greatest common factor of 12x4, 18x3, and 30x2.

Solution

To find the GCF of 12x4, 18x3, and 30x2, we note that:

  1. The greatest common factor of 12, 18, and 30 is 6.
  2. The monomials 12x4, 18x3, and 30x2 have the variable x in common.
  3. The highest power of x in common is x2.

Thus, the greatest common factor is GCF(12x4,18x3,30x2)=6x2. Note what happens when we write each of the given monomials as a product of the greatest common factor and a second monomial:

12x4=6x22x218x3=6x23x30x2=6x25

Observe that the set of second monomial factors (2x2, 3x, and 5) contain no additional common factors.

You Try 2.7.5

Find the greatest common factor of 6y3, 15y2, and 9y5.

Answer

3y2

Factor Out the GCF

Earlier, we multiplied a monomial and polynomial by distributing the monomial times each term in the polynomial.

2x(3x2+4x7)=2x3x2+2x4x2x7=6x3+8x214x

In this section we reverse that multiplication process using the distributive property, ab+ac=a(b+c)

We present you with the final product and ask you to bring back the original multiplication problem. In the case 6x3+8x214x, the greatest common factor of 6x3, 8x2, and 14x is 2x. We then use the distributive property to factor out 2x from each term of the polynomial.

6x3+8x214x=2x3x2+2x4x2x7=2x(3x2+4x7)

Factoring

Factoring is the process of writing an expression as a product of polynomials.

Let’s look at a few examples that factor out the GCF.

Example 2.7.6

Factor: 6x2+10x+14

Solution

The greatest common factor (GCF) of 6x2, 10x and 14 is 2. Factor out the GCF.

6x2+10x+14x=23x2+25x+27=2(3x2+5x+7)

The factored form of 6x2+10x+14 is 2(3x2+5x+7).

Check:

Check that you factored correctly by multiplying:

2(3x2+5x+7)=23x2+25x+27=6x2+10x+14

That’s the original polynomial, so we factored correctly.

You Try 2.7.6

Factor: 9y215y+12

Answer

3(3y25y+4)

Example 2.7.7

Factor: 12y532y4+8y2

Solution

The greatest common factor (GCF) of 12y5, 32y4 and 8y2 is 4y2. Factor out the GCF.

12y532y4+8y2=4y23y34y28y2+4y22=4y2(3y38y2+2)

Check: Multiply. Distribute the monomial 4y2.

4y2(3y38y2+2)=4y23y34y28y2+4y22=12y532y4+8y2

That’s the original polynomial. We have factored correctly.

You Try 2.7.7

Factor: 8x6+20x424x3

Answer

4x3(2x3+5x6)

Example 2.7.8

Factor: 12a3b+24a2b2+12ab3

Solution

The greatest common factor (GCF) of 12a3b, 24a2b2 and 12ab3 is 12ab. Factor out the GCF.

12a3b+24a2b2+12ab3=12aba212ab2ab+12abb2=12ab(a2+2ab+b2)

Check: Multiply. Distribute the monomial 12ab.

12ab(a2+2ab+b2)=12aba212ab2ab+12abb2=12a3b+24a2b2+12ab3

That’s the original polynomial. We have factored correctly.

You Try 2.7.8

Factor: 15s2t4+6s3t2+9s2t2

Answer

3s2t2(5t2+2s+3)

Remember that the distributive property allows us to pull the GCF out in front of the expression or to pull it out in back. In symbols:

ab+ac=a(b+c) or ba+ca=(b+c)a

We can also use the distributive property to factor out binomial factors.

Example 2.7.9

Factor: 2x(3x+2)+5(3x+2)

Solution

In this case, the greatest common factor (GCF) is 3x+2.

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

Because of the commutative property of multiplication, it is equally valid to pull the GCF out in front.

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

Note that the order of factors differs from the first solution, but because of the commutative property of multiplication, the order does not matter. The answers are the same.

You Try 2.7.9

Factor: 3x2(4x7)+8(4x7)

Answer

(3x2+8)(4x7)

Example 2.7.10

Factor: 15a(a+b)12(a+b)

Solution

In this case, the greatest common factor (GCF) is 3(a+b).

15a(a+b)12(a+b)=3(a+b)5a3(a+b)4=3(a+b)(5a4)

Alternate solution:

It is possible that you might not notice that 15 and 12 are divisible by 3, factoring out only the common factor a+b.

15a(a+b)12(a+b)=15a(a+b)12(a+b)=(15a12)(a+b)

However, you now need to notice that you can continue to factor by factoring out 3 from both 15a and 12.

=3(5a4)(a+b)

Note that the order of factors differs from the first solution, but because of the commutative property of multiplication, the order does not matter. The answers are the same.

You Try 2.7.10

Factor: 24m(m2n)+20(m2n)

Answer

4(6m+5)(m2n)

Factoring by Grouping

The last factoring skill in this section involves four-term expressions. The technique for factoring a four-term expression is called factoring by grouping.

Example 2.7.11

Factor by grouping: x2+8x+3x+24

Solution

We “group” the first and second terms, noting that we can factor an x out of both of these terms. Then we “group” the third and fourth terms, noting that we can factor 3 out of both of these terms.

x2+8x+3x+24=x(x+8)+3(x+8)

Now we can factor x+8 out of both of these terms.

(x+3)(x+8)

You Try 2.7.11

Factor by grouping: x26x+2x12

Answer

(x+2)(x6)

Let’s try a grouping that contains some negative signs.

Example 2.7.12

Factor by grouping: x2+4x7x28

Solution

We “group” the first and second terms, noting that we can factor x out of both of these terms. Then we “group” the third and fourth terms, then try to factor a 7 out of both these terms.

x2+4x7x28=x(x+4)+7(x4)

 

This does not lead to a common factor. Let’s try again, this time factoring a 7 out of the third and fourth terms.

x2+4x7x28=x(x+4)7(x+4)

 

That worked! We now factor out a common factor x+4.

(x7)(x+4)

It is important to be careful with negatives when factoring by grouping. In the previous example, it would be incorrect to write x2+4x7x28 as (x2+4x)(7x28) because if you distribute the negative through, the last term is +28 which is not the original polynomial.

You Try 2.7.12

Factor by grouping: x25x4x+20

Answer

(x4)(x5)

Let’s increase the size of the numbers a bit.

Example 2.7.13

Factor by grouping: 6x28x+9x12

Solution

Note that we can factor 2x out of the first two terms and 3 out of the second two terms.

x2+4x9x12=2x(3x4)7(3x4)

 

Now we have a common factor 3x4 which we can factor out.

(2x+3)(3x4)

You Try 2.7.13

Factor by grouping: 15x2+9x+10x+6

Answer

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

As the numbers get larger and larger, you need to factor out the (GCF) from each grouping. If not, you won’t get a common factor to finish the factoring.

Example 2.7.14

Factor by grouping: 24x232x45x+60

Solution

Suppose that we factor 8x out of the first two terms and 5 out of the second two terms.

24x232x45x+60=8x(3x4)5(9x12)

 

That did not work, as we don’t have a common factor to complete the factoring process. However, note that we can still factor out a 3 from 9x12. As we’ve already factored out a 5, and now we see can factor out an additional 3, this means that we should have factored out 3 times 5, or 15, to begin with. Let’s start again, only this time we’ll factor 15 out of the second two terms.

24x232x45x+60=8x(3x4)15(3x4)

Beautiful! We can now factor out 3x4.

(8x15)(3x4)

You Try 2.7.14

Factor by grouping: 36x284x+15x35

Answer

(12x+5)(3x7)


This page titled 2.7: The Greatest Common Factor and Factor by Grouping is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Katherine Skelton.

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