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6.1 The Greatest Common Factor; Factoring by Grouping
1
Find the greatest common factor of a list of terms.
2
Factor out the greatest common factor.
3
Factor by grouping.
The Greatest Common Factor: Factoring by Grouping
Factoring is the opposite of multiplying. For example,
Multiplying
Factoring
6 · 2 = 12
12 = 6 · 2
Factors
Product
Product
other factored forms of 12 are
− 6(−2), 3 · 4, −3(−4), 12 · 1,
and
Factors
−12(−1).
More than two factors may be used, so another factored form of 12 is
2 · 2 · 3. The positive integer factors of 12 are
1, 2, 3, 4, 6, 12.
Slide 6.1-3
Objective 1
Find the greatest common factor
of a list of terms.
Slide 6.1-4
Find the greatest common factor of a list of terms.
An integer that is a factor of two or more integers is called a common
factor of those integers. For example, 6 is a common factor of 18 and
24. Other common factors of 18 and 24 are 1, 2, and 3. The greatest
common factor (GCF) of a list of integers is the largest common
factor of those integers. Thus, 6 is the greatest common factor of 18
and 24.
Factoring numbers into prime factors is the first step in finding the
greatest common factor of a list of numbers.
Slide 6.1-5
Find the greatest common factor of a list of terms.
(cont’d)
Factors of a number are also divisors of the number. The greatest
common factor is actually the same as the greatest common
divisor. The are many rules for deciding what numbers to divide into
a given number. Here are some especially useful divisibility rules for
small numbers.
Slide 6.1-6
Find the greatest common factor of a list of terms.
(cont’d)
Finding the Greatest Common Factor (GCF)
Step 1: Factor. Write each number in prime factored form.
Step 2: List common factors. List each prime number or each
variable that is a factor of every term in the list.
Step 3: Choose least exponents. Use as exponents on the common
prime factors the least exponent from the prime factored
forms.
Step 4: Multiply. Multiply the primes from Step 3. If there are no
primes left after Step 3, the greatest common factor is 1.
Slide 6.1-7
EXAMPLE 1 Finding the Greatest Common Factor for Numbers
Find the greatest common factor for each list of numbers.
Solution:
50, 75
GCF = 25
12, 18, 26, 32
GCF = 2
50  2  5  5
75  3  5  5
12  2  2  3
26  2 13
18  2  3  3
32  2  2  2  2  2
22  2 11
24  2  2  2  3
22, 23, 24
GCF = 1
23  1 23
Slide 6.1-8
Find the greatest common factor of a list of terms.
(cont’d)
The GCF can also be found for a list of variable terms. The exponent
on a variable in the GCF is the least exponent that appears in all the
common factors.
Slide 6.1-9
EXAMPLE 2 Finding the Greatest Common Factor for Variable Terms
Find the greatest common factor for each list of terms.
Solution:
16r 9 , 10r15 , 8r12
16r 9  1 2  2  2  2  r 9
GCF = 2r 9
10r15  1 2  5  r15
8r12  2  2  2  r12
s 4t 5 , s3t 6 , s9t 2
s 4t 5  s 4  t 5
GCF = s 3t 2
s 3t 6  s 3  t 6
s 9t 2  s 9  t 2
Slide 6.1-10
Objective 2
Factor out the greatest common
factor.
Slide 6.1-11
Factor out the greatest common factor.
Writing a polynomial (a sum) in factored form as a product is called
factoring. For example, the polynomial
3m + 12
has two terms: 3m and 12. The GCF of these terms is 3. We can write
3m + 12 so that each term is a product of 3 as one factor.
3m + 12 = 3 · m + 3 · 4
= 3(m + 4)
GCF = 3
Distributive property
The factored form of 3m + 12 is 3(m + 4). This process is called
factoring out the greatest common factor.
The polynomial 3m + 12 is not in factored form when written as 3 · m + 3 · 4.
The terms are factored, but the polynomial is not. The factored form of 3m +12 is
the product 3(m + 4).
Slide 6.1-12
EXAMPLE 3 Factoring Out the Greatest Common Factor
Write in factored form by factoring out the greatest common factor.
Solution:
 6x 2  x 2  2 
6 x 4  12 x 2
30t  25t  10t
6
5
r r
12
 5t 4  6t 2  5t  2 
4
r
10
10
r
2
 1
8 p q  16 p q  12 p q  4 p q  2 p  4 p q  3q
5
2
6 3
4
7
4
2
2
5

Be sure to include the 1 in a problem like r12 + r10. Always check that the
factored form can be multiplied out to give the original polynomial.
Slide 6.1-13
EXAMPLE 4 Factoring Out the Greatest Common Factor
Write in factored form by factoring out the greatest common factor.
Solution:
6 p  q  r  p  q
 p  q  6  r 
y 4  y  3  4  y  3
 y  3  y 4  4 
Slide 6.1-14
Objective 3
Factor by grouping.
Slide 6.1-15
Factor by grouping.
When a polynomial has four terms, common factors can
sometimes be used to factor by grouping.
Factoring a Polynomial with Four Terms by Grouping
Step 1: Group terms. Collect the terms into two groups so that each
group has a common factor.
Step 2: Factor within groups. Factor out the greatest common factor
from each group.
Step 3: Factor the entire polynomial. Factor out a common binomial
factor from the results of Step 2.
Step 4: If necessary, rearrange terms. If Step 2 does not result in a
common binomial factor, try a different grouping.
Slide 6.1-16
EXAMPLE 5 Factoring by Grouping
Factor by grouping.
Solution:
pq  5q  2 p  10
q  p  5  2  p  5   p  5 q  2
2 xy  3 y  2 x  3
y  2 x  3  1 2 x  3   2 x  3 y  1
2a  4a  3ab  6b
2
2a  a  2  3b  a  2   a  2 2a  3b 
x3  3x 2  5 x  15
x2  x  3  5  x  3
  x  3  x 2  5 
Slide 6.1-17
EXAMPLE 6 Rearranging Terms before Factoring by Grouping
Factor by grouping.
Solution:
6 y 2  20w  15 y  8 yw
 6 y 2  15 y  20w  8 yw
 3 y  2 y  5   4w  2 y  5 
  2 y  5 3 y  4w
9mn  4  12m  3n
 9mn  12m  3n  4
 3m  3n  4   1 3n  4 
  3m  1 3n  4
Slide 6.1-18