Chapter 7 Section 1 Common Factors

1
Chapter 7 Section 1Common Factors

• Objective 1 To factor a monomial from a
• polynomial.
• Objective 2 To factor by grouping.

2
Factors

• Factors are the numbers that you multiply to get
  another number.

• The factors of the number 12 are

Lets write those numbers in order.
3
Factors

• Factors are the numbers that you multiply to get
  another number.

• The factors of the number 12 are

1 , 2 , 3 , 4 , 6 , 12
4
Greatest Common Factor

• The greatest common factor (GCF) of two numbers
  is the largest factor of both numbers.

• Lets find the GCF of 24 and 36.

First find the factors of 24 and 36.
5
Greatest Common Factor

• The greatest common factor (GCF) of two numbers
  is the largest factor of both numbers.

• Lets find the GCF of 24 and 36.

Factors of 24 1, 2, 3, 4, 6, 8, 12, 24 Factors
of 36 1, 2, 3, 4, 6, 9, 12, 18, 36
What is the LARGEST factor of both numbers?
6
Greatest Common Factor

• The greatest common factor (GCF) of two numbers
  is the largest factor of both numbers.

• Lets find the GCF of 24 and 36.

Factors of 24 1, 2, 3, 4, 6, 8, 12, 24 Factors
of 36 1, 2, 3, 4, 6, 9, 12, 18, 36
The GCF is 12.
7
Greatest Common Factor - Variables

• The greatest common factor (GCF) of two variable
  expressions is each variable that appears in both
  expressions with the smallest exponent of each
  variable.

• Lets find the GCF of x5y3z8 and x2y9z.

Write each variable with the smallest exponent.
8
Greatest Common Factor - Variables

• The greatest common factor (GCF) of two variable
  expressions is each variable that appears in both
  expressions with the smallest exponent of each
  variable.

• Lets find the GCF of x5y3z8 and x2y9z.

9
Greatest Common Factor - Variables

• The greatest common factor (GCF) of two variable
  expressions is each variable that appears in both
  expressions with the smallest exponent of each
  variable.

• Lets find the GCF of x5y3z8 and x2y9z.

The GCF is x2y3z
10
Greatest Common Factor - Variables

• The greatest common factor (GCF) of two variable
  expressions is each variable that appears in both
  expressions with the smallest exponent of each
  variable.

• Lets find the GCF of a10b2c4 and a3b5c2d.

Write each variable with the smallest exponent.
11
Greatest Common Factor - Variables

• The greatest common factor (GCF) of two variable
  expressions is each variable that appears in both
  expressions with the smallest exponent of each
  variable.

• Lets find the GCF of a10b2c4 and a3b5c2d.

12
Greatest Common Factor - Variables

• The greatest common factor (GCF) of two variable
  expressions is each variable that appears in both
  expressions with the smallest exponent of each
  variable.

• Lets find the GCF of a10b2c4 and a3b5c2d.

13
Greatest Common Factor - Variables

• The greatest common factor (GCF) of two variable
  expressions is each variable that appears in both
  expressions with the smallest exponent of each
  variable.

• Lets find the GCF of a10b2c4 and a3b5c2d.

14
Greatest Common Factor - Variables

• The greatest common factor (GCF) of two variable
  expressions is each variable that appears in both
  expressions with the smallest exponent of each
  variable.

• Lets find the GCF of a10b2c4 and a3b5c2d.

The d is not in both expressions, so it is NOT a
common factor.
15
Greatest Common Factor - Variables

• The greatest common factor (GCF) of two variable
  expressions is each variable that appears in both
  expressions with the smallest exponent of each
  variable.

• Lets find the GCF of a10b2c4 and a3b5c2d.

The GCF is a3b2c2
16
Find the GCF Example 1
First, find the GCF of the numbers.
17
Find the GCF Example 1
18
Find the GCF Example 1
Now, find the GCF of the variables.
19
Find the GCF Example 1
20
Find the GCF Example 1
The GCF is 2x2y
21
Find the GCF Example 2
First, find the GCF of the numbers.
22
Find the GCF Example 2
23
Find the GCF Example 2
24
Find the GCF Example 2
25
Find the GCF Example 2
Only one c. It is NOT a common factor.
26
Find the GCF Example 2
The GCF is 6a2b
27
Objective 1

• To factor a monomial from a polynomial

28
Factoring a polynomial The Steps

• Find the GCF of the terms of the polynomial.
• Write the GCF and parenthesis.
• Fill in the parenthesis by dividing.

29
Factoring a polynomial Example 3
Find the GCF of the three terms.
30
Factoring a polynomial Example 3
Write a pair of parentheses beside the GCF.
31
Factoring a polynomial Example 3
What do we multiply by 5x to get 5x3?
Subtract the exponents.
32
Factoring a polynomial Example 3
What do we multiply by 5x to get -35x2?
33
Factoring a polynomial Example 3
What do we multiply by 5x to get 10x?
34
Factoring a polynomial Example 3
Factoring is like the reverse of the Distributive
Property
35
Factoring a polynomial Example 3
36
Factoring a polynomial Example 4
Find the GCF of the three terms.
37
Factoring a polynomial Example 4
Write a pair of parentheses beside the GCF.
38
Factoring a polynomial Example 4
What do we multiply by 3xy2 to get 21x2y2?
39
Factoring a polynomial Example 4
What do we multiply by 3xy2 to get -6xy5?
40
Factoring a polynomial Example 4
What do we multiply by 3xy2 to get 15x4y2?
41
Factoring a polynomial Example 4
42
Factoring a polynomial Example 5
Find the GCF of the two terms.
43
Factoring a polynomial Example 5
Write Parentheses.
44
Factoring a polynomial Example 5
45
Factoring a polynomial Example 5
46
Factoring a polynomial Example 6
Find the GCF of the three terms.
47
Factoring a polynomial Example 6
Write Parentheses.
48
Factoring a polynomial Example 6
49
Factoring a polynomial Example 6
50
Factoring a polynomial Example 7
Find the GCF of the three terms.
51
Factoring a polynomial Example 7
Write Parentheses.
52
Factoring a polynomial Example 7
53
Factoring a polynomial Example 7
54
Objective 2

• To factor by grouping

55
Factor By Grouping The Steps

• Find the common binomial.
• Write two pairs of parentheses.
• Put common binomial in one pair of parentheses.
• Put whatevers left in other pair of parenthesis.

56
Factor By Grouping Example 8
Look for a binomial that appears twice.
57
Factor By Grouping Example 8
Write two pairs of parentheses.
58
Factor By Grouping Example 8
Write the common binomial in the first
parentheses.
59
Factor By Grouping Example 8
Write whats left in the other parentheses.
60
Factor By Grouping Example 8
61
Factor By Grouping Example 9
Look for a binomial that appears twice.
62
Factor By Grouping Example 9
Write two pairs of parentheses.
63
Factor By Grouping Example 9
Write the common binomial in the first
parentheses.
64
Factor By Grouping Example 9
Write whats left in the other parentheses.
65
Factor By Grouping Example 9
66
Factor By Grouping Example 10
Look for a binomial that appears twice.
67
Factor By Grouping Example 10
Write two pairs of parentheses.
68
Factor By Grouping Example 10
Write the common binomial in the first
parentheses.
69
Factor By Grouping Example 10
Write whats left in the other parentheses.
70
Factor By Grouping Example 10
71
Factor By Grouping The Steps

• Find the common binomial.
• Write two pairs of parentheses.
• Put common binomial in one pair of parentheses.
• Put whatevers left in other pair of parenthesis.

Sometimes, you must rewrite the expression to
find the common binomial.
Lets practice rewriting some expressions.
72
Rewriting an Expression
Suppose we need the binomial (b a) If can
change positive 1 to negative 1, We can write (b
a).
73
Rewriting an Expression
74
Rewriting an Expression
Suppose we need the binomial (x 5) If can
change positive 2 to negative 2, We can write (x
5).
75
Rewriting an Expression
76
Factor By Grouping Example 11
Look for a binomial that appears twice.
77
Factor By Grouping Example 11
We need to change (y x) to (x y). If we
change positive 5 to negative 5, We can change (y
x) to (x y)
78
Factor By Grouping Example 11
Look for a binomial that appears twice.
79
Factor By Grouping Example 11
80
Factor By Grouping Example 11
Write two pairs of parentheses.
81
Factor By Grouping Example 11
Write the common binomial in the first
parentheses.
82
Factor By Grouping Example 11
Write whats left in the other parentheses.
83
Factor By Grouping Example 11
84
Factor By Grouping Example 12
Look for a binomial that appears twice.
85
Factor By Grouping Example 12
We need to change (2 x) to (x 2). If we
change positive y to negative y, We can change (2
x) to (x 2)
86
Factor By Grouping Example 12
Look for a binomial that appears twice.
87
Factor By Grouping Example 12
88
Factor By Grouping Example 12
Write two pairs of parentheses.
89
Factor By Grouping Example 12
Write the common binomial in the first
parentheses.
90
Factor By Grouping Example 12
Write whats left in the other parentheses.
91
Factor By Grouping Example 12
92
Factor By Grouping Example 13
Look for a binomial that appears twice.
93
Factor By Grouping Example 13
We need to change (2 5x) to (5x 2). If we
change negative 3 to positive 3, We can change (2
5x) to (5x 2)
94
Factor By Grouping Example 13
Look for a binomial that appears twice.
95
Factor By Grouping Example 13
Write two pairs of parentheses.
96
Factor By Grouping Example 13
Write the common binomial in the first
parentheses.
97
Factor By Grouping Example 13
Write whats left in the other parentheses.
98
Factor By Grouping Example 13
99
Factor By Grouping The Steps

• Find the common binomial.
• Write two pairs of parentheses.
• Put common binomial in one pair of parentheses.
• Put whatevers left in other pair of parenthesis.

Sometimes, you must group the terms in a
polynomial in order to find a common binomial.
Lets look at some examples like that.
100
Factor By Grouping Example 14
Group the first two terms together and the last
two terms together.
101
Factor By Grouping Example 14
Now, take out the GCF in each group.
102
Factor By Grouping Example 14
103
Factor By Grouping Example 14
104
Factor By Grouping Example 14
Look for a binomial that appears twice.
105
Factor By Grouping Example 14
Write two pairs of parentheses.
106
Factor By Grouping Example 14
Write the common binomial in the first
parentheses.
107
Factor By Grouping Example 14
Write whats left in the other parentheses.
108
Factor By Grouping Example 14
109
Factor By Grouping Example 15
Group the first two terms together and the last
two terms together.
110
Factor By Grouping Example 15
Take out the GCF in each group.
111
Factor By Grouping Example 15
Look for a binomial that appears twice.
112
Factor By Grouping Example 15
Write two pairs of parentheses.
113
Factor By Grouping Example 15
Write the common binomial in the first
parentheses.
114
Factor By Grouping Example 15
Write whats left in the other parentheses.
115
Factor By Grouping Example 15
116
Factor By Grouping Example 16
Group the first two terms together and the last
two terms together.
117
Factor By Grouping Example 16
Take out the GCF in each group.
118
Factor By Grouping Example 16
Look for a binomial that appears twice.
119
Factor By Grouping Example 16
Write two pairs of parentheses.
120
Factor By Grouping Example 16
Write the common binomial in the first
parentheses.
121
Factor By Grouping Example 16
Write whats left in the other parentheses.
122
Factor By Grouping Example 16
123
STOP!
Be Careful! If the 3rd term is negative, you
must change the sign of the 4th term when using
this method of factoring by grouping.
124
Last 3 examples
The 3rd term is positive.
125
Factor By Grouping Example 17
The 3rd term is negative.
When we group, we must change the sign of the 4th
term.
126
Factor By Grouping Example 17
Group the 1st two terms.
127
Factor By Grouping Example 17
Group the last terms. Change the sign of the 4th
term.
128
Factor By Grouping Example 17
Take out the GCF in each group.
129
Factor By Grouping Example 17
Look for a binomial that appears twice.
130
Factor By Grouping Example 17
Write two pairs of parentheses.
131
Factor By Grouping Example 17
Write the common binomial in the first
parentheses.
132
Factor By Grouping Example 17
Write whats left in the other parentheses.
133
Factor By Grouping Example 17
134
Factor By Grouping Example 18
The 3rd term is negative.
When we group, we must change the sign of the 4th
term.
135
Factor By Grouping Example 18
Group the 1st two terms.
136
Factor By Grouping Example 18
Group the last terms. Change the sign of the 4th
term.
137
Factor By Grouping Example 18
Take out the GCF in each group.
138
Factor By Grouping Example 18
Look for a binomial that appears twice.
139
Factor By Grouping Example 18
Write the common binomial in the first
parentheses.
140
Factor By Grouping Example 18
Write whats left in the other parentheses.
141
Factor By Grouping Example 18
142
Factor By Grouping Example 19
The 3rd term is negative.
When we group, we must change the sign of the 4th
term.
143
Factor By Grouping Example 19
Group the 1st two terms.
144
Factor By Grouping Example 19
Group the last terms. Change the sign of the 4th
term.
145
Factor By Grouping Example 19
Take out the GCF in each group.
146
Factor By Grouping Example 19
Take out the GCF in each group.
147
Factor By Grouping Example 19
Look for binomial that appears twice.
148
Factor By Grouping Example 19
If we change negative 4 to positive 4, We can
write x 3.
149
Factor By Grouping Example 19
150
Factor By Grouping Example 19
Look for a binomial that appears twice.
151
Factor By Grouping Example 19
152
Factor By Grouping Example 19
153
HOMEWORK Page 405 406

• Complete problems 3 67 ODD.
• Make sure you highlight your answers BEFORE you
  come to class.
• Check your answers.