Using the Greatest Common Factor and Factor by Grouping

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 \(3 x\).

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

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.?

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

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.

	\(5 q(q+7)-6(q+7)\)
Factor the GCF, (q + 7).	\((q+7)(5 q-6)\)
Check on your own by multiplying.

Try It 7.27

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

Try It 7.28

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