Transcript Factor out the greatest common factor.

Chapter 6
Section 1
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6.1
1
2
3
The Greatest Common Factor;
Factoring by Grouping
Find the greatest common factor of a list
of terms.
Factor out the greatest common factor.
Factor by grouping.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Find the greatest common factor of a list
of terms.
Recall from Chapter 1 that to factor means “to write a
quantity as a product.” 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,
Factors
and
−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.
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Slide 6.1 - 3
Objective 1
Find the greatest common factor
of a list of terms.
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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.
Recall from Chapter 1 that a prime number has only itself
and 1 as factors. In Section 1.1, we factored numbers into
prime factors. This is the first step in finding the greatest
common factor of a list of numbers.
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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.
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Slide 6.1 - 6
Find the greatest common factor of a list of
terms. (cont’d)
Find the greatest common factor (GCF) of a list of numbers as
follows.
Step 1: Factor. write each number in a prime factored form.
Step 2: List common factors. List each prime number that
is a factor of every number in the list. (If a prime
does not appear in one of the prime factored forms,
it cannot appear in the greatest common factor.)
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.
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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  2  5  5
50, 75
75  3  5  5
GCF = 25
12, 18, 26, 32
12  2  2  3
26  2 13
GCF = 2
18  2  3  3
32  2  2  2  2  2
12, 13, 14
12  2  2  3
GCF = 1
13  1 13
14  2  7
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Slide 6.1 - 8
EXAMPLE 2
Finding the Greatest Common
Factor for Variable Terms
Find the greatest common factor for each list of terms.
16r 9 , 10r15 , 8r12
GCF = 2r 9
s 4t 6 , s3t 6 , s9t 2
GCF = s3t 2
 x2 y3 ,  xy5
GCF = xy3 or  xy3
Solution:
16r 9  1 2  2  2  2  r 9
10r15  1 2  5  r15
8r12  2  2  2  r12
s 4t 6  s 4  t 6
s3t 6  s3  t 6
s 9t 2  s 9  t 2
 x2 y3  1 x2  y3
 xy5  1 x  y5
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Slide 6.1 - 9
Objective 2
Factor out the greatest common
factor.
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Slide 6.1 - 10
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
Distributive Property.
= 3(m + 4)
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).
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Slide 6.1 - 11
EXAMPLE 3
Factoring Out the Greatest
Common Factor
Factor out the GCF. In the fifth example, use fractions
in the factored form. Solution:
4
2
6 x  12 x
 6x 2  x 2  2 
 5t 4  6t 2  5t  2 
30t 6  25t 5  10t 4
r r
12
 r 10  r 2  1
10
8 p q  16 p q 12 p q
5 2
6 3
1 9 3 2
x  x
4
4
4 7
 4 p q  2 p  4 p q  3q
4
2
2
5

1 2 7
 x  x  3
4
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.
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Slide 6.1 - 12
EXAMPLE 4
Factoring Out the Greatest
Common Factor
Factor out the greatest common factor.
6 p  q  r  p  q
Solution:
 p  q  6  r 
y4  y  3  4  y  3
4
y

3
y

   4
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Slide 6.1 - 13
Objective 3
Factor by grouping.
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Slide 6.1 - 14
Factor by grouping.
When a polynomial has four terms, common factors can
sometimes be used to factor 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.
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Slide 6.1 - 15
EXAMPLE 5
Factoring by Grouping
Factor by grouping.
pq  5q  2 p  10
Solution: q  p  5  2  p  5   p  5 q  2
2 xy  3 y  2 x  3
y  2x  3  1 2x  3   2x  3 y 1
2a  4a  3ab  6b
2
2a  a  2  3b  a  2   a  2 2a  3b
x3  3x 2  5x  15
x2  x  3  5  x  3
  x  3  x 2  5 
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Slide 6.1 - 16
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
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Slide 6.1 - 17