Using the Greatest Common Factor and Factor by Grouping

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.

Example 7.1

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

Find the GCF of 54 and 36.

Solution

Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form.	Factor \(\underline{54}\) and \(\underline{36}\).
Step 2. In each column, circle the common factors.	Circle the 2,3, and 3 that are shared by both numbers.
Step 3. Bring down the common factors that all expressions share.	Bring down the 2,3, and 3 and then multiply.	\(GCF=2 \cdot \qquad 3 \cdot 3\)
Step 4. Multiply the factors.		\(GCF=18 \) 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.

\(54=18 \cdot 3\)
\(36=18 \cdot 2\)

Try It 7.1

Find the GCF of 48 and 80.

Try It 7.2

Find the GCF of 18 and 40.

HOW TO

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

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

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.	\(\mathrm{GCF}=3 \cdot 3 \cdot \ qquad x \cdot x \cdot x\)
Multiply the factors.	\(\mathrm{GCF}=9 x^{3}\)
	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.	\(\mathrm{GCF}=2 \cdot \qquad x \cdot \quad y\)
Multiply the factors.	\(GCF=2xy\)
	The GCF of \(4 x^{2} y\) and \(6 x y^{3}\) is \(2 \mathrm{xy}\).

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.	\(G C F=3 \cdot \qquad \quad x\)
Multiply the factors.	\(\mathrm{GCF}=3x\)
	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: \(14 x^{3}, 70 x^{2}, 105 x\).

Distributive Property

If a,b,c are real numbers, then

\(a(b+c)=a b+a c \quad\) and \(\quad a b+a c=a(b+c)\)

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

Example 7.5

How to Factor the Greatest Common Factor from a Polynomial

Factor: \(4x+12\).

Solution

Step 1. Find the GCF of all the terms of the polynomial.	Find the GCF of \(4 x\) and 12 .
Step 2. Rewrite each term as a product using the GCF.	Rewrite \(4 x\) and 12 as products of their GCF, 4. \( \begin{array}{l} 4 x=4 \cdot x \\ 12=4 \cdot 3 \end{array} \)	\( \begin{array}{c} 4 x+12 \\ 4 \cdot x+4 \cdot 3 \end{array} \)
Step 3. Use the "reverse" Distributive Property to factor the expression.		\(4(x+3)\)
Step 4. Check by multiplying the factors.		\( \begin{array}{c} 4(x+3) \\ 4 \cdot x+4 \cdot 3 \\ 4 x+12 \text{✓} \end{array} \)

Try It 7.9

Factor: \(6a+24\).

Try It 7.10

Factor: \(2b+14\).

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.

Noun: \(7\) is a \(factor\) of \(14\)

Verb: \(factor\) 3 from \(3 a+3\)

Example 7.6

Factor: \(5a+5\).

Solution

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

Try It 7.11

Factor: \(14 x+14\).

Try It 7.12

Factor: \(12 p+12\).

Example 7.7

Factor: \(12x−60\).

Solution

Find the GCF of \(12x\) and \(60\).
	\(12 x-60\)
Rewrite each term as a product using the GCF.	\(12 \cdot x-12 \cdot 5\)
Factor the GCF.	\(12(x-5)\)
Check by multiplying the factors.
\(12(x−5)\)
\(12 \cdot x-12 \cdot 5\)
\(12x−60✓\)

Try It 7.13

Factor: \(18u−36\).

Try It 7.14

Factor: \(30y−60\).

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

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

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 \(3 x\).

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

Try It 7.21

Factor: \(9 x y^{2}+6 x^{2} y^{2}+21 y^{3}\).

Try It 7.22

Factor: \(3 p^{3}-6 p^{2} q+9 p q^{3}\).

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 \(8 y\) and 24 is 8 . Since the expression \(-8 y-24\) has a negative leading coefficient, we use \(-8\) as the GCF.
Rewrite each term using the GCF.	\(-8 y-24\) \(-8 \cdot y+(-8) \cdot 3\)
Factor the GCF.	\(-8(y+3)\)
Check.
\(-8(y+3)\)
\(-8 \cdot y+(-8) \cdot 3\)
\(-8 y-24 \text{✓}\)

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.?

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.

Try It 7.27

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

Try It 7.28

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

Example 7.15

How to Factor by Grouping

Factor: \(xy+3y+2x+6\).

Solution

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.

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

Example 7.16

Factor: \(x^{2}+3 x-2 x-6\).

Solution

Try It 7.31

Factor: \(x^{2}+2 x-5 x-10\).

Try It 7.32

Factor: \(y^{2}+4 y-7 y-28\).