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Factor: 9 x 2 + 7 x .

x (9 x + 7)

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Factor: 5 a 2 12 a .

a (5 a − 12)

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When there are several common factors, as we’ll see in the next two examples, good organization and neat work helps!

Factor: 4 x 3 20 x 2 .

Solution

.
.
Rewrite each term. .
Factor the GCF. .
Check. .
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Factor: 2 x 3 + 12 x 2 .

2 x 2 ( x + 6)

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Factor: 6 y 3 15 y 2 .

3 y 2 (2 y − 5)

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Factor: 21 y 2 + 35 y .

Solution

Find the GCF of 21 y 2 and 35 y .
.
Rewrite each term. .
Factor the GCF. .
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Factor: 18 y 2 + 63 y .

9 y (2 y + 7)

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Factor: 32 k 2 + 56 k .

8 k (4 k + 7)

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Factor: 14 x 3 + 8 x 2 10 x .

Solution

Previously, we found the GCF of 14 x 3 , 8 x 2 , and 10 x to be 2 x .

14 x 3 + 8 x 2 10 x
Rewrite each term using the GCF, 2x. .
Factor the GCF. 2 x ( 7 x 2 + 4 x 5 )
.
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Factor: 18 y 3 6 y 2 24 y .

6 y (3 y 2 y − 4)

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Factor: 16 x 3 + 8 x 2 12 x .

4 x (4 x 2 + 2 x − 3)

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

Factor: −9 y 27 .

Solution

When the leading coefficient is negative, the GCF will be negative. Ignoring the signs of the terms, we first find the GCF of 9 y and 27 is 9. .
Since the expression −9y−27 has a negative leading coefficient, we use −9 as the GCF.
9 y 27
Rewrite each term using the GCF. .
Factor the GCF. 9 ( y + 3 )
.
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Factor: −5 y 35 .

−5( y + 7)

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Factor: −16 z 56 .

−8(2 z + 7)

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Pay close attention to the signs of the terms in the next example.

Factor: −4 a 2 + 16 a .

Solution

The leading coefficient is negative, so the GCF will be negative.
.
Since the leading coefficient is negative, the GCF is negative, −4 a .
−4 a 2 + 16 a
Rewrite each term. .
Factor the GCF. 4 a ( a 4 )
Check on your own by multiplying.
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Factor: −7 a 2 + 21 a .

−7 a ( a − 3)

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Factor: −6 x 2 + x .

x (6 x − 1)

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Key concepts

  • Find the greatest common factor.
    1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
    2. List all factors—matching common factors in a column. In each column, circle the common factors.
    3. Bring down the common factors that all expressions share.
    4. Multiply the factors.
  • Distributive Property
    • If a , b , c are real numbers, then
      a ( b + c ) = a b + a c and a b + a c = a ( b + c )
  • Factor the greatest common factor from a polynomial.
    1. Find the GCF of all the terms of the polynomial.
    2. Rewrite each term as a product using the GCF.
    3. Use the Distributive Property ‘in reverse’ to factor the expression.
    4. Check by multiplying the factors.

Practice makes perfect

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

18 a , 6 a 2 , 22 a 3

2 a

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40 y , 10 y 2 , 90 y 3

10 y

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35 x 3 , 10 x 4 , 5 x 5

5 x 3

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14 b 2 , 35 b 3 , 63 b 4

7 b 2

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Factor the Greatest Common Factor from a Polynomial

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

3 n 2 + 21 n + 12

3( n 2 + 7 n + 4)

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6 q 2 + 30 q + 42

6( q 2 + 5 q + 7)

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9 c 2 + 22 c

c (9 c + 22)

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17 x 2 + 7 x

x (17 x + 7)

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3 r 2 + 27 r

3 r ( r + 9)

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30 u 2 10 u

10 u (3 u − 1)

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55 y 11 y 4

11 y (5 − y 3 )

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45 c 3 15 c 2

15 c 2 (3 c − 1)

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6 c 3 6 c d 2

6 c ( c 2 d 2 )

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48 x 3 + 72 x 2

24 x 2 (2 x + 3)

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Practice Key Terms 1

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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