Greatest Common Factor and Factor by Grouping

7.1 Greatest Common Factor and Factor by Grouping

Learning Objectives

By the end of this section, you will be able to:

Find the greatest common factor of two or more expressions
Factor the greatest common factor from a polynomial
Factor by grouping

Be Prepared 7.1

Before you get started, take this readiness quiz.

Factor 56 into primes.
If you missed this problem, review Example 1.7.

Be Prepared 7.2

Find the least common multiple of 18 and 24.
If you missed this problem, review Example 1.10.

Be Prepared 7.3

Simplify $−3 (6 a + 11)$ .
If you missed this problem, review Example 1.135.

Find the Greatest Common Factor of Two or More Expressions

Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.

This figure has two factors being multiplied. They are 8 and 7. Beside this equation there are other factors multiplied. They are 2x and (x+3). The product is given as 2x^2 plus 6x. Above the figure is an arrow towards the right with multiply inside. Below the figure is an arrow to the left with factor inside.

We have learned how to factor numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM.

Greatest Common Factor

The greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.

First we’ll find the GCF of two numbers.

Example 7.1

How to Find the Greatest Common Factor of Two or More Expressions

Find the GCF of 54 and 36.

Solution

This table has three columns. In the first column are the steps for factoring. The first row has the first step, factor each coefficient into primes and write all variables with exponents in expanded form. The second column in the first row has “factor 54 and 36”. The third column in the first row has 54 and 36 factored with factor trees. The prime factors of 54 are circled and are 3, 3, 2, and3. The prime factors of 36 are circled and are 2,3,2,3.

The second row has the second step of “in each column, circle the common factors. The second column in the second row has the statement “circle the 2, 3 and 3 that are shared by both numbers”. The third column in the second row has the prime factors of 36 and 54 in rows above each other. The common factors of 2, 3, and 3 are circled.

The third row has the step “bring down the common factors that all expressions share”. The second column in the third row has “bring down the 2,3, and 3 then multiply”. The third column in the third row has “GCF = 2 times 3 times 3”.

The fourth row has the fourth step “multiply the factors”. The second column in the fourth row is blank. The third column in the fourth row has “GCF = 18” and “the GCF of 54 and 36 is 18”.

Notice that, because the GCF is a factor of both numbers, 54 and 36 can be written as multiples of 18.

\begin{matrix} 54 = 18 \cdot 3 \\ 36 = 18 \cdot 2 \end{matrix}

Try It 7.1

Find the GCF of 48 and 80.

Try It 7.2

Find the GCF of 18 and 40.

We summarize the steps we use to find the GCF below.

How To

Find the Greatest Common Factor (GCF) of two expressions.

Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
Step 2. List all factors—matching common factors in a column. In each column, circle the common factors.
Step 3. Bring down the common factors that all expressions share.
Step 4. Multiply the factors.

In the first example, the GCF was a constant. In the next two examples, we will get variables in the greatest common factor.

Example 7.2

Find the greatest common factor of $27 x^{3} and 18 x^{4}$ .

Solution

Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column.
Bring down the common factors.
Multiply the factors.
	The GCF of $27 x^{3}$ and $18 x^{4}$ is $9 x^{3} .$

Try It 7.3

Find the GCF: $12 x^{2}, 18 x^{3}$ .

Try It 7.4

Find the GCF: $16 y^{2}, 24 y^{3}$ .

Example 7.3

Find the GCF of $4 x^{2} y, 6 x y^{3}$ .

Solution

Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column.
Bring down the common factors.
Multiply the factors.
	The GCF of $4 x^{2} y$ and $6 x y^{3}$ is $2 xy$ .

Try It 7.5

Find the GCF: $6 a b^{4}, 8 a^{2} b$ .

Try It 7.6

Find the GCF: $9 m^{5} n^{2}, 12 m^{3} n$ .

Example 7.4

Find the GCF of: $21 x^{3}, 9 x^{2}, 15 x$ .

Solution

Factor each coefficient into primes and write the variables with exponents in expanded form. Circle the common factors in each column.
Bring down the common factors.
Multiply the factors.
	The GCF of $21 x^{3}$ , $9 x^{2}$ and $15 x$ is $3 x .$

Try It 7.7

Find the greatest common factor: $25 m^{4}, 35 m^{3}, 20 m^{2}$ .

Try It 7.8

Find the greatest common factor: $14 x^{3}, 70 x^{2}, 105 x$ .

Factor the Greatest Common Factor from a Polynomial

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, 12 as $2 \cdot 6 or 3 \cdot 4),$ in algebra, it can be useful to represent a polynomial in factored form. One way to do this is by finding the GCF of all the terms. Remember, we multiply a polynomial by a monomial as follows:

\begin{matrix} 2 (x + 7) & factors \\ 2 \cdot x + 2 \cdot 7 \\ 2 x + 14 & product \end{matrix}

Now we will start with a product, like $2 x + 14$ , and end with its factors, $2 (x + 7)$ . To do this we apply the Distributive Property “in reverse.”

We state the Distributive Property here just as you saw it in earlier chapters and “in reverse.”

Distributive Property

If $a, b, c$ are real numbers, then

\begin{matrix} a (b + c) = a b + a c & and & a b + a c = a (b + c) \end{matrix}

The form on the left is used to multiply. The form on the right is used to factor.

So how do you use the Distributive Property to factor a polynomial? You just find the GCF of all the terms and write the polynomial as a product!

Example 7.5

How to Factor the Greatest Common Factor from a Polynomial

Factor: $4 x + 12$ .

Solution

This table has three columns. In the first column are the steps for factoring. The first row has the first step, “Find the G C F of all the terms of the polynomial”. The second column in the first row has “find the G C F of 4 x and 12”. The third column in the first row has 4 x factored as 2 times 2 times x and below it 18 factored as 2 times 2 times 3. Then, below the factors are the statements, “G C F = 2 times 2” and “G C F = 4”.

The second row has the second step “rewrite each term as a product using the G C F”. The second column in the second row has the statement “Rewrite 4 x and 12 as products of their G C F, 4” Then the two equations 4 x = 4 times x and 12 = 4 times 3. The third column in the second row has the expressions 4x + 12 and below this 4 times x + 4 times 3.

The third row has the step “Use the reverse distributive property to factor the expression”. The second column in the third row is blank. The third column in the third row has “4(x + 3)”.

The fourth row has the fourth step “check by multiplying the factors”. The second column in the fourth row is blank. The third column in the fourth row has three expressions. The first is 4(x + 3), the second is 4 times x + 4 times 3. The third is 4 x + 12.

Try It 7.9

Factor: $6 a + 24$ .

Try It 7.10

Factor: $2 b + 14$ .

How To

Factor the greatest common factor from a polynomial.

Step 1. Find the GCF of all the terms of the polynomial.
Step 2. Rewrite each term as a product using the GCF.
Step 3. Use the “reverse” Distributive Property to factor the expression.
Step 4. Check by multiplying the factors.

Factor as a Noun and a Verb

We use “factor” as both a noun and a verb.

Example 7.6

Factor: $5 a + 5$ .

Solution

Find the GCF of 5a and 5.

Rewrite each term as a product using the GCF.
Use the Distributive Property "in reverse" to factor the GCF.
Check by mulitplying the factors to get the orginal polynomial.
$5 (a + 1)$
$5 \cdot a + 5 \cdot 1$
$5 a + 5 ✓$

Try It 7.11

Factor: $14 x + 14$ .

Try It 7.12

Factor: $12 p + 12$ .

The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.

Example 7.7

Factor: $12 x - 60$ .

Solution

Find the GCF of 12x and 60.

Rewrite each term as a product using the GCF.
Factor the GCF.
Check by mulitplying the factors.
$12 (x - 5)$
$12 \cdot x - 12 \cdot 5$
$12 x - 60 ✓$

Try It 7.13

Factor: $18 u - 36$ .

Try It 7.14

Factor: $30 y - 60$ .

Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.

Example 7.8

Factor: $4 y^{2} + 24 y + 28$ .

Solution

We start by finding the GCF of all three terms.

Find the GCF of $4 y^{2}$ , $24 y$ and 28.

Rewrite each term as a product using the GCF.
Factor the GCF.
Check by mulitplying.
$4 (y^{2} + 6 y + 7)$
$4 \cdot y^{2} + 4 \cdot 6 y + 4 \cdot 7$
$4 y^{2} + 24 y + 28 ✓$

Try It 7.15

Factor: $5 x^{2} - 25 x + 15$ .

Try It 7.16

Factor: $3 y^{2} - 12 y + 27$ .

Example 7.9

Factor: $5 x^{3} - 25 x^{2}$ .

Solution

Find the GCF of $5 x^{3}$ and $25 x^{2} .$

Rewrite each term.
Factor the GCF.
Check.
$5 x^{2} (x - 5)$
$5 x^{2} \cdot x - 5 x^{2} \cdot 5$
$5 x^{3} - 25 x^{2} ✓$

Try It 7.17

Factor: $2 x^{3} + 12 x^{2}$ .

Try It 7.18

Factor: $6 y^{3} - 15 y^{2}$ .

Example 7.10

Factor: $21 x^{3} - 9 x^{2} + 15 x$ .

Solution

In a previous example we found the GCF of $21 x^{3}, 9 x^{2}, 15 x$ to be $3 x$ .


Rewrite each term using the GCF, 3x.
Factor the GCF.
Check.
$3 x (7 x^{2} - 3 x + 5)$
$3 x \cdot 7 x^{2} - 3 x \cdot 3 x + 3 x \cdot 5$
$21 x^{3} - 9 x^{2} + 15 x ✓$

Try It 7.19

Factor: $20 x^{3} - 10 x^{2} + 14 x$ .

Try It 7.20

Factor: $24 y^{3} - 12 y^{2} - 20 y$ .

Example 7.11

Factor: $8 m^{3} - 12 m^{2} n + 20 m n^{2}$ .

Solution

Find the GCF of $8 m^{3}$ , $12 m^{2} n$ , $20 m n^{2}$ .

Rewrite each term.
Factor the GCF.
Check.
$4 m (2 m^{2} - 3 m n + 5 n^{2})$
$4 m \cdot 2 m^{2} - 4 m \cdot 3 m n + 4 m \cdot 5 n^{2}$
$8 m^{3} - 12 m^{2} n + 20 m n^{2} ✓$

Try It 7.21

Factor: $9 x y^{2} + 6 x^{2} y^{2} + 21 y^{3}$ .

Try It 7.22

Factor: $3 p^{3} - 6 p^{2} q + 9 p q^{3}$ .

When the leading coefficient is negative, we factor the negative out as part of the GCF.

Example 7.12

Factor: $−8 y - 24$ .

Solution

When the leading coefficient is negative, the GCF will be negative.

Ignoring the signs of the terms, we first find the GCF of 8y and 24 is 8. Since the expression −8y − 24 has a negative leading coefficient, we use −8 as the GCF.
Rewrite each term using the GCF.
Factor the GCF.
Check.
$−8 (y + 3)$
$−8 \cdot y + (−8) \cdot 3$
$−8 y - 24 ✓$

Try It 7.23

Factor: $−16 z - 64$ .

Try It 7.24

Factor: $−9 y - 27$ .

Example 7.13

Factor: $−6 a^{2} + 36 a$ .

Solution

The leading coefficient is negative, so the GCF will be negative?

Since the leading coefficient is negative, the GCF is negative, −6a.
Rewrite each term using the GCF.
Factor the GCF.
Check.
$−6 a (a - 6)$
$−6 a \cdot a + (−6 a) (−6)$
$−6 a^{2} + 36 a ✓$

Try It 7.25

Factor: $−4 b^{2} + 16 b$ .

Try It 7.26

Factor: $−7 a^{2} + 21 a$ .

Example 7.14

Factor: $5 q (q + 7) - 6 (q + 7)$ .

Solution

The GCF is the binomial $q + 7$ .


Factor the GCF, (q + 7).
Check on your own by multiplying.

Try It 7.27

Factor: $4 m (m + 3) - 7 (m + 3)$ .

Try It 7.28

Factor: $8 n (n - 4) + 5 (n - 4)$ .

Factor by Grouping

When there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts.

(Not all polynomials can be factored. Just like some numbers are prime, some polynomials are prime.)

Example 7.15

How to Factor by Grouping

Factor: $x y + 3 y + 2 x + 6$ .

Solution

This table gives the steps for factoring x y + 3 y + 2 x + 6. In the first row there is the statement, “group terms with common factors”. In the next column, there is the statement of no common factors of all 4 terms. The last column shows the first two terms grouped and the last two terms grouped.

The second row has the statement, “factor out the common factor from each group”. The second column in the second row states to factor out the GCF from the two separate groups. The third column in the second row has the expression y(x + 3) + 2(x + 3).

The third row has the statement, “factor the common factor from the expression”. The second column in this row points out there is a common factor of (x + 3). The third column in the third row shows the factor of (x + 3) factored from the two groups, (x + 3) times (y + 2).

The last row has the statement, “check”. The second column in this row states to multiply (x + 3)(y + 2). The product is shown in the last column of the original polynomial x y + 3 y + 2 x + 6.

Try It 7.29

Factor: $x y + 8 y + 3 x + 24$ .

Try It 7.30

Factor: $a b + 7 b + 8 a + 56$ .

How To

Factor by grouping.

Step 1. Group terms with common factors.
Step 2. Factor out the common factor in each group.
Step 3. Factor the common factor from the expression.
Step 4. Check by multiplying the factors.

Example 7.16

Factor: $x^{2} + 3 x - 2 x - 6$ .

Solution

$\begin{matrix} There is no GCF in all four terms. & x^{2} + 3 x −2 x - 6 \\ Separate into two parts. & \underset{⎵}{x^{2} + 3 x} \underset{⎵}{−2 x - 6} \\ \begin{matrix} Factor the GCF from both parts. Be careful \\ with the signs when factoring the GCF from \\ the last two terms. \end{matrix} & \begin{matrix} x (x + 3) - 2 (x + 3) \\ (x + 3) (x - 2) \end{matrix} \\ Check on your own by multiplying. \end{matrix}$

Try It 7.31

Factor: $x^{2} + 2 x - 5 x - 10$ .

Try It 7.32

Factor: $y^{2} + 4 y - 7 y - 28$ .

Media

Access these online resources for additional instruction and practice with greatest common factors (GFCs) and factoring by grouping.

Section 7.1 Exercises

Practice Makes Perfect

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

8, 18

24, 40

72, 162

150, 275

10a, 50

5b, 30

$3 x, 10 x^{2}$

$21 b^{2}, 14 b$

$8 w^{2}, 24 w^{3}$

10.

$30 x^{2}, 18 x^{3}$

11.

$10 p^{3} q, 12 p q^{2}$

12.

$8 a^{2} b^{3}, 10 a b^{2}$

13.

$12 m^{2} n^{3}, 30 m^{5} n^{3}$

14.

$28 x^{2} y^{4}, 42 x^{4} y^{4}$

15.

$10 a^{3}, 12 a^{2}, 14 a$

16.

$20 y^{3}, 28 y^{2}, 40 y$

17.

$35 x^{3}, 10 x^{4}, 5 x^{5}$

18.

$27 p^{2}, 45 p^{3}, 9 p^{4}$

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

19.

$4 x + 20$

20.

$8 y + 16$

21.

$6 m + 9$

22.

$14 p + 35$

23.

$9 q + 9$

24.

$7 r + 7$

25.

$8 m - 8$

26.

$4 n - 4$

27.

$9 n - 63$

28.

$45 b - 18$

29.

$3 x^{2} + 6 x - 9$

30.

$4 y^{2} + 8 y - 4$

31.

$8 p^{2} + 4 p + 2$

32.

$10 q^{2} + 14 q + 20$

33.

$8 y^{3} + 16 y^{2}$

34.

$12 x^{3} - 10 x$

35.

$5 x^{3} - 15 x^{2} + 20 x$

36.

$8 m^{2} - 40 m + 16$

37.

$12 x y^{2} + 18 x^{2} y^{2} - 30 y^{3}$

38.

$21 p q^{2} + 35 p^{2} q^{2} - 28 q^{3}$

39.

$−2 x - 4$

40.

$−3 b + 12$

41.

$5 x (x + 1) + 3 (x + 1)$

42.

$2 x (x - 1) + 9 (x - 1)$

43.

$3 b (b - 2) - 13 (b - 2)$

44.

$6 m (m - 5) - 7 (m - 5)$

Factor by Grouping

In the following exercises, factor by grouping.

45.

$x y + 2 y + 3 x + 6$

46.

$m n + 4 n + 6 m + 24$

47.

$u v - 9 u + 2 v - 18$

48.

$p q - 10 p + 8 q - 80$

49.

$b^{2} + 5 b - 4 b - 20$

50.

$m^{2} + 6 m - 12 m - 72$

51.

$p^{2} + 4 p - 9 p - 36$

52.

$x^{2} + 5 x - 3 x - 15$

Mixed Practice

In the following exercises, factor.

53.

$−20 x - 10$

54.

$5 x^{3} - x^{2} + x$

55.

$3 x^{3} - 7 x^{2} + 6 x - 14$

56.

$x^{3} + x^{2} - x - 1$

57.

$x^{2} + x y + 5 x + 5 y$

58.

$5 x^{3} + 3 x^{2} - 5 x - 3$

Everyday Math

59.

Area of a rectangle The area of a rectangle with length 6 less than the width is given by the expression $w^{2} - 6 w$ , where $w =$ width. Factor the greatest common factor from the polynomial.

60.

Height of a baseball The height of a baseball t seconds after it is hit is given by the expression $−16 t^{2} + 80 t + 4$ . Factor the greatest common factor from the polynomial.

Writing Exercises

61.

The greatest common factor of 36 and 60 is 12. Explain what this means.

62.

What is the GCF of $y^{4}, y^{5}, and y^{10}$ ? Write a general rule that tells you how to find the GCF of $y^{a}, y^{b}, and y^{c}$ .

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has the following statements all to be preceded by “I can…”. The first is “find the greatest common factor of two or more expressions”. The second is “factor the greatest common factor from a polynomial”. The third is “factor by grouping”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential—every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.