gcf present distributive property in reverse
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ac bc2 2a b
3 3a b
3 3a b
ac bc2 2a b
3 3a b
3 3a b
GCF present
ac bc
c a b
ac cb
Distributive Property in Reverse
ac bc
c a b
ac cb
Distributive Property in Reverse
2 11m m
11m m
Distributive Property in Reverse
Example 1
m
2 11m m
11m m
Distributive Property in Reverse
Example 1
ac bc2 2a b
3 3a b
3 3a b
c a b
ac bc2 2a b
3 3a b
3 3a b
c a b
Difference of 2 SQUARES
2 2a b
2 36x Example 1
2 0 36x x 2 6 66 3xx x
6 6x x
21) 9x
22) 16y
23) 4 9x
24) 9x
21) 9x
22) 16y
23) 4 9x
24) 9x
3 3x x
4 4y y
2 3 2 3x x
21) 9x
22) 16y
23) 4 9x
3 3x x
4 4y y
2 3 2 3x x
2 2a b a b a b
2 2a b a b a b
2 36x
Example 1
2 36x
2 2a b
x x 6 62 36x
2 2a b
( )( )
x x 6 62 36x
a b a b
2 2a b
( )( )x x6 6
2 36x x x 6 6
a b a b
2 2a b
( )( )x x6 6
2 36x x x 6 6
2x
( )( )x x6 6
2 36x x x 6 6
2x 6x
( )( )x x6 6
2 36x x x 6 6
2x 6x 6x
( )( )x x6 6
2 36x x x 6 6
2x 6x 6x 36
( )( )x x6 6
2 36x x x 6 6
2x 6x 6x 36
( )( )x x6 6
2 36x
24 1x
Example 2
24 1x
2 2a b
2 2x x 1 124 1x
2 2a b
( )( )
24 1x 2 2x x 1 1
2 2a b
a b a b
( )( )2x 2x1 1
24 1x 2 2x x 1 1
2 2a b
a b a b
( )( )2x 2x1 1
24 1x 2 2x x 1 1
( )( )2x 2x1 1
24 1x 2 2x x 1 1
2 25x
Example 3 2 2a b
6 36x
Example 4
6 36x
2 2a b
6 36x
6x
2 2a bIs this a perfect
Square?
3 3x x 6 66 36x
2 2a b
( )( )
6 66 36x 3 3x x
2 2a b
a b a b
( )( )3x 3x6 6
6 66 36x 3 3x x
2 2a b
a b a b
( )( )3x 3x6 6
6 66 36x 3 3x x
( )( )3x 3x6 6
6 66 36x 3 3x x
ac bc2 2a b
3 3a b
3 3a b
c a b
a b a b
Difference of 2 CUBES
3 3a b
The Difference of Two Cubes:
V length width height
The difference of cubes can be modeled using volume.
The volume of a cube is:
V l w h
Imagine taking a cube with a volume of x3
x
x
x3
V x x xor
V x
The volume of the resulting figure
must be
x3 - y3. x
x
x
yy
y
and cutting out and removing a smaller cube with a volume of y3 .
Imagine taking a cube with a volume of x3
We call this the difference of two cubes.
x
x
x
yy
y
and cutting out and removing a smaller cube with a volume of y3 .
Imagine taking a cube with a volume of x3
The volume of the resulting figure
must be
x3 - y3.
We are going to discover how to factor the difference of two cubes by breaking the diagram into rectangular prisms and finding the sum of the volumes of the rectangular prisms.
xx
x - y
2
V x x x y
x x yx
xx
yy
y
xx
x - y
2
V x x x y
x x yx
xx
yy
y
xx
x - y
2
V x x x y
x x yx
xx
yy
y
y
x-yx
V x y x y
xy x y
xx
x - y
2
V x x x y
x x yx
xx
yy
y
y
x-yx
V x y x y
xy x y
xx
x - y
2
V x x x y
x x yx
xx
yy
y
y
x-yx
V x y x y
xy x y
yx-yy
2
V y y x y
y x y
xx
x - y
2
V x x x y
x x yx
xx
yy
y
y
x-yx
V x y x y
xy x y
yx-yy
2
V y y x y
y x y
3 3 x y 2 x x y xy x y 2 y x y
3 3 x y
x3 y3 x2 x y xy x y y2 x y
3 3 x y 2 x x y xy x y 2 y x y
3 3 x y
x3 y3 x2 x y xy x y y2 x y
3 3 x y 2 x x y xy x y 2 y x y
3 3 x y
x3 y3 x2 x y xy x y y2 x y
3 3 x y 2 x x y xy x y 2 y x y
3 3 x y
x3 y3 x2 x y xy x y y2 x y
3 3 x y 2 x x y xy x y 2 y x y
3 3 x y
x3 y3 x2 x y xy x y y2 x y x y x2 xy y2
3 3 x y
x3 y3 x2 x y xy x y y2 x y
x3 y3 x y x2 xy y2
x y x2 xy y2 3 3 x y
x3 y3 x y x2 xy y2
3 3 2 2 x y x y x xy y
ac bc2 2a b
3 3a b
3 3a b
c a b
a b a b
2 2a b a ab b
3 3a b 2 2b a ba ab
38 27x
3 32 3x
3 3a b 2 2b a ba ab
38 27x
3 32 3x
3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x
3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x 3b
3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x 3b 222 22 3 3 3x xx
3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x 3b 222 22 3 3 3x xx
3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x 3b 222 22 3 3 3x xx
3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x 3b 222 22 3 3 3x xx
3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x 3b 222 22 3 3 3x xx
3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x 3b 222 22 3 3 3x xx
3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x 3b 222 22 3 3 3x xx
22 3 4 6 9x x x
3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x 3b 222 22 3 3 3x xx
22 3 4 6 9x x x
3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x 3b 222 22 3 3 3x xx
22 3 4 6 9x x x
3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x 3b 222 22 3 3 3x xx
22 3 4 6 9x x x
38 27x 22 3 4 6 9x x x
4x2 6x 9
2x
–3
38x 212x 18x212x 18x 27
38 27x 22 3 4 6 9x x x
4x2 6x 9
2x
–3
38x 212x 18x212x 18x 27
327 1x
327 1x
23 1 9 3 1x x x
9x2 3x 1
3x
–1
327x 29x 3x29x 3x 1
327 1x
23 1 9 3 1x x x
9x2 3x 1
3x
–1
327x 29x 3x29x 3x 1
ac bc2 2a b
3 3a b
3 3a b
c a b
a b a b
2 2a b a ab b
ac bc2 2a b
3 3a b
3 3a b
c a b
a b a b
2 2a b a ab b
Sum of 2 CUBES
ac bc2 2a b
3 3a b
3 3a b
c a b
a b a b
2 2a b a ab b
2 2a b a ab b
3 3a b
2 2a b a ab b
3 3a b 2 2b a ba ab
3 8x 3 32x
3 3a b 2 2b a ba ab
3 8x 3 32x
a x
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
22 2 4x x x
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
22 2 4x x x
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
22 2 4x x x
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
22 2 4x x x
3 8x 22 2 4x x x
x2 –2x 4
x
2
3x 22x 4x22x 4x 8
3 8x 22 2 4x x x
x2 –2x 4
x
2
3x 22x 4x22x 4x 8
3 27x
3 27x
23 3 9x x x
ac bc2 2a b
3 3a b
3 3a b
GCF present
Difference of 2 SQUARES
Difference of 2 CUBES
Sum of 2 CUBES
ac bc2 2a b
3 3a b
3 3a b
c a b
a b a b
2 2a b a ab b
2 2a b a ab b
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