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FACTORING
RULES
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*GCF( Greatest Common Factor) – First Rule
4 TERMS
Grouping 3 TERMS
Perfect Square Trinomial
AC Method with Grouping
2 TERMS
Difference Of Two Squares
Sum or Difference Of Two Cubes
2 2 2
2 2 2
a 2ab b (a b)
a 2ab b (a b)
2 2a b (a b)(a b)
3 3 2 2
3 3 2 2
a b (a b)(a ab b )
a b (a b)(a ab b )
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GCFGreatest Common Factor
First Rule to Always Check
3
2
2
1) 3
3
3
y y
y
y
y y
y
3
2 2
2
22) 8 16
8 2
2
8
8
a a a
a
aa
a
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3 2 3
3 2 3
3 2 3
4) 12 16 48
3 4 14 4 2
34 4 12
4
p p t t
p p t t
p p t t
23) 2
2
2
a
ab a ax
b aa ax
b a xa
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2 2 2 2
2
5) 4 2 4 2
2 2
2 2
a b c d
a b c d
a b c d
2 2
2 2
2 2
1 1
3 3
1 16)
3
3
1
3R h r h
R r
R
h h
h r
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7) 3
3
4 4
4
x xx
xx
8) 2 7
2
1 1
1 7
y y
yy
y
29)
1
1
a b a b
a
a b a b
a b
a b
a b
b
a b
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4 TERMS - Grouping
2
1 2
1
2
) 2 2
[ 2
2
2
]
Group GroupGCF GCF
GCFx y
a
x y ax ay
x y x y
x y x y
x
a
y
a
a
a
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2
3 2
3
16
2
2
1 2
2
2
2) 2 4 32 64
2 [ 2 16 32]
2 2 2
2 2
16
16
Group GroupGCxF GCF
GCFx
x z x z xz z
z x x x
z x
x
x
z x
x
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3 TERMS
1) Perfect Square Trinomials
2) AC Method With Grouping
We will explore factoring trinomials using the ac method with grouping next
and come back to Perfect Square Trinomials later.
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Factoring Trinomials
by
Using The
AC Method
With
Grouping
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4 3 26 14 40y y y
2 22 2[ 7 02 2 23 2 ]y yyy y
The first rule of factoring is to factor out the Greatest Common Factor
(GCF).
Factor the trinomial completely.
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Stop! Check that you have factored the (GCF) correctly by distributing it
back through the remaining polynomial to obtain the original
trinomial.
2 2[3 20]2 7 y yy2 22 2[ 7 02 2 23 2 ]y yyy y
4 3 26 14 40y y y
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2ax cbx
To factor , we must find two integers whose product is -60 and whose sum is 7.
To factor , we must find two integers whose product is ac and whose sum is b.
After factoring out the (GCF), the remaining polynomial is of the form
4 3 26 14 40y y y
22 3 20[ ]2 7yy y 2ax cbx
2 73 20y y
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Key number 60
6012 12( 5) 5 7
FACTORS OF 60 SUM OF FACTORS OF 60
22 3 20[ ]2 7yy y
1( 60) 60 1 ( 60) 59
2( 30) 60 2 ( 30) 28
3( 20) 60 3 ( 20) 17
4( 15) 60 4 ( 15) 11
5( 12) 60 5 ( 12) 7
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ac = b = 7
Replace b = 7 in our original expression with
b = 12 + (-5).
7 0]2y
12 y 5 0]2y
60
22 3 20[ ]2 7yy y
6012 12( 5) 5 7
2 22 [3y y2 22 [3y y
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FINISH FACTORING BY GROUPING
22 3 20[ ]2 7yy y
2
3
2
Group 1 Group 2G
5CF GCF
2 012 5 ]3[ 2
y
y yy y
2
3GCF C
5G F
[ ( 4) ( 45 )]2 3
y
yy y y
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1722 ( 4)(3 5)y y y
FACTORED COMPLETELY
4 3 26 14 40y y y 22 3 20[ ]2 7yy y
GC
2
F( 4)
2 [ ( 4) (5 ]3 4)
y
y yy y
2
3
2
Group 1 Group 2G
5CF GCF
2 012 5 ]3[ 2
y
y yy y
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Practice Problems
2
2 2
2
2
2 2
2
1) 12 4 16
2) 6 29 28
3) 8 30 18
4) 3 10 8
5) 10 7 12
6) 6 3 18
a a
a ab b
x x
h h
m mn n
y y
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SUM OF FACTORS OF
GCF
FACTORS OF
KEY #
21) 12 4 16 a a
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SUM OF FACTORS OF
GCF
FACTORS OF
KEY #
2 22) 6 29 28 a ab b
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SUM OF FACTORS OF
GCF
FACTORS OF
KEY #
23) 8 30 18x x
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SUM OF FACTORS OF
GCF
FACTORS OF
KEY #
24) 3 10 8 h h
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SUM OF FACTORS OF
GCF
FACTORS OF
KEY #
2 25) 10 7 12 m mn n
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SUM OF FACTORS OF
GCF
FACTORS OF
KEY #
26) 6 3 18y y
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Answers To Practice Problems
1) 4(3 4)( 1)
2) (3 4 )(2 7 )
3) 2(4 3)( 3)
4) 1(3 2)( 4)
5) (5 4 )(2 3 )
6) 3(2 3)( 2)
a a
a b a b
x x
h h
m n m n
y y
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Perfect Square Trinomials
2 2
22 2
2 2
2
1) 25 70 49
5 2 5 7 7
5 7
2
ba
m mn n
m
a ab
m n n
m
b a
n
b
22 2
22 2
2
2
a ab b a b
a ab b a b
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4 3 2
2
2 2
22
2
22
2 2
2) 4 20 25
4 20 25
2 2 2 5
2
5
2 5
a b
a ab b a b
x x x
x x x
x x x
x x
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2 TERMS
1) Difference of Two Squares
2) Sum and Difference of Two Cubes
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Difference of Two Squares
2 2a b a b a b
2
2 2
1) 9
3 3 3
x
x x x
2
2
2 2
2) 2 200
2 100
2 10 2 10 10
p
p
p p p
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4
2 22 2 2
2
3) 81
9 9 9
9 3 3
x
x x x
x x x
2
2
22
544) 6
259
625
3 3 36 6
5 5 5
t
t
t t t
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Sum and Difference of Two Cubes
3 3 2 2
3 3 2 2
a b a b a ab b
a b a b a ab b
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3
3
3
3
3 2
3
2
2
1) 125
125
5
5 5 25
x
x
x
x x
a b a b a ab b
x
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4 3
3 3
3 3
3 3 2
2 2
2
2) 16 128
16 8
16 2
16 2 2 4
a b a b
r rs
r
a ab
r s
r r s
r r s r rs s
b
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3
3
3
3
3 2
3
2
2
3) 216
216
6
6 6 36
x
x
x
x x
a b a b a ab b
x
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3 3 2
3 3
3 3
3 3
2
2
2
4) 64 8
8 8
8 2
8 2 4 2
a b a b a ab
m x n x
x m n
x m n
x m n m n
b
m n
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What purpose does factoring serve?
Factoring is an algebraic process which allows us to solve quadratic equations pertaining to real-world applications, such as remodeling a
kitchen or building a skyscraper.
We will cover the concept of solving quadratic equations and then investigate some real-
world applications.
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Solving Quadratic Equations
A quadratic equation is an equation that can be written in standard form
where a, b, and c represent real numbers, and
2 0ax bx c
0a
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We will solve some quadratic equations
using factoring and the
Zero-Factor Property.
When the product of two real numbers is 0, at least one of them is 0.
If a and b represent real numbers, and
if then a=0 or b=00ab
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Solve Each Equation
1) 3 2 0
3 0 and 2 0
3 2
x x
x x
x x
2) 7 3 10 0
7 0 and 3 10 0
100
3
a a
a a
a a
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2
2
3) 9 3 3 25
9 27 3 25
9 30 25 0
3 5 3 5 0
3 5 0
5
3
a a a
a a a
a a
a a
a
a
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2
2
2
4) 8 3 30
3 8 24 30
5 24 30
5 6 0
2 3 0
2 0 and 3 0
2 3
n n
n n n
n n
n n
n n
n n
n n
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3 2
2
5) 3 2 0
3 2 0
1 2 0
0, 1 0, and 2 0
0, 1, 2
x x x
x x x
x x x
x x x
x
3
2
6) 6 6 0
6 1 0
6 1 1 0
6 0, 1 0, 1 0
0, 1, 1
n n
n n
n n n
n n n
n
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REAL-WORLDAPPLICATIONS
USINGQUADRATICEQUATIONS
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The height h in feet reached by a dolphin t seconds afterbreaking the surface of the water is given by hHow long will it take the dolphin to jump out of the water and touch the trainer’s hand?
216 32t t
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From the top of the building a ball is thrown straight up with an initial velocity of 32 feet per second. The equation below gives the height s of the ball t seconds after thrown. Find the maximum height reached by the ball and the time it takes for the ball to hit the ground.
216 32 48s t t