gcf present distributive property in reverse
TRANSCRIPT
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ac bc2 2a b
3 3a b
3 3a b
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ac bc2 2a b
3 3a b
3 3a b
GCF present
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ac bc
c a b
ac cb
Distributive Property in Reverse
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ac bc
c a b
ac cb
Distributive Property in Reverse
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2 11m m
11m m
Distributive Property in Reverse
Example 1
m
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2 11m m
11m m
Distributive Property in Reverse
Example 1
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ac bc2 2a b
3 3a b
3 3a b
c a b
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ac bc2 2a b
3 3a b
3 3a b
c a b
Difference of 2 SQUARES
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2 2a b
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2 36x Example 1
2 0 36x x 2 6 66 3xx x
6 6x x
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21) 9x
22) 16y
23) 4 9x
24) 9x
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21) 9x
22) 16y
23) 4 9x
24) 9x
3 3x x
4 4y y
2 3 2 3x x
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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
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2 2a b a b a b
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2 36x
Example 1
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2 36x
2 2a b
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x x 6 62 36x
2 2a b
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( )( )
x x 6 62 36x
a b a b
2 2a b
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( )( )x x6 6
2 36x x x 6 6
a b a b
2 2a b
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( )( )x x6 6
2 36x x x 6 6
2x
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( )( )x x6 6
2 36x x x 6 6
2x 6x
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( )( )x x6 6
2 36x x x 6 6
2x 6x 6x
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( )( )x x6 6
2 36x x x 6 6
2x 6x 6x 36
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( )( )x x6 6
2 36x x x 6 6
2x 6x 6x 36
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( )( )x x6 6
2 36x
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24 1x
Example 2
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24 1x
2 2a b
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2 2x x 1 124 1x
2 2a b
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( )( )
24 1x 2 2x x 1 1
2 2a b
a b a b
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( )( )2x 2x1 1
24 1x 2 2x x 1 1
2 2a b
a b a b
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( )( )2x 2x1 1
24 1x 2 2x x 1 1
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( )( )2x 2x1 1
24 1x 2 2x x 1 1
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2 25x
Example 3 2 2a b
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6 36x
Example 4
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6 36x
2 2a b
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6 36x
6x
2 2a bIs this a perfect
Square?
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3 3x x 6 66 36x
2 2a b
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( )( )
6 66 36x 3 3x x
2 2a b
a b a b
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( )( )3x 3x6 6
6 66 36x 3 3x x
2 2a b
a b a b
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( )( )3x 3x6 6
6 66 36x 3 3x x
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( )( )3x 3x6 6
6 66 36x 3 3x x
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ac bc2 2a b
3 3a b
3 3a b
c a b
a b a b
Difference of 2 CUBES
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3 3a b
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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
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Imagine taking a cube with a volume of x3
x
x
x3
V x x xor
V x
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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
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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.
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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.
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xx
x - y
2
V x x x y
x x yx
xx
yy
y
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xx
x - y
2
V x x x y
x x yx
xx
yy
y
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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
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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
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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
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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
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3 3 x y 2 x x y xy x y 2 y x y
3 3 x y
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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
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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
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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
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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
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x3 y3 x2 x y xy x y y2 x y x y x2 xy y2
3 3 x y
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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
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x3 y3 x y x2 xy y2
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3 3 2 2 x y x y x xy y
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ac bc2 2a b
3 3a b
3 3a b
c a b
a b a b
2 2a b a ab b
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3 3a b 2 2b a ba ab
38 27x
3 32 3x
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3 3a b 2 2b a ba ab
38 27x
3 32 3x
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3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x
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3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x 3b
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3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x 3b 222 22 3 3 3x xx
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3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x 3b 222 22 3 3 3x xx
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3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x 3b 222 22 3 3 3x xx
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3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x 3b 222 22 3 3 3x xx
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3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x 3b 222 22 3 3 3x xx
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3 3a b 2 2b a ba ab
38 27x
3 32 3x
2a x 3b 222 22 3 3 3x xx
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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
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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
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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
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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
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38 27x 22 3 4 6 9x x x
4x2 6x 9
2x
–3
38x 212x 18x212x 18x 27
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38 27x 22 3 4 6 9x x x
4x2 6x 9
2x
–3
38x 212x 18x212x 18x 27
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327 1x
![Page 87: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/87.jpg)
327 1x
23 1 9 3 1x x x
9x2 3x 1
3x
–1
327x 29x 3x29x 3x 1
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327 1x
23 1 9 3 1x x x
9x2 3x 1
3x
–1
327x 29x 3x29x 3x 1
![Page 89: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/89.jpg)
![Page 90: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/90.jpg)
ac bc2 2a b
3 3a b
3 3a b
c a b
a b a b
2 2a b a ab b
![Page 91: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/91.jpg)
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
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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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3 3a b
2 2a b a ab b
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3 3a b 2 2b a ba ab
3 8x 3 32x
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3 3a b 2 2b a ba ab
3 8x 3 32x
a x
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3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b
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3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
![Page 98: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/98.jpg)
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
![Page 99: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/99.jpg)
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
![Page 100: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/100.jpg)
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
![Page 101: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/101.jpg)
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
![Page 102: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/102.jpg)
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
![Page 103: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/103.jpg)
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
![Page 104: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/104.jpg)
3 3a b 2 2b a ba ab
3 8x 3 32x
a x 2b 2 22 2 2x x x
![Page 105: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/105.jpg)
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
![Page 106: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/106.jpg)
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
![Page 107: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/107.jpg)
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
![Page 108: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/108.jpg)
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
![Page 109: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/109.jpg)
3 8x 22 2 4x x x
x2 –2x 4
x
2
3x 22x 4x22x 4x 8
![Page 110: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/110.jpg)
3 8x 22 2 4x x x
x2 –2x 4
x
2
3x 22x 4x22x 4x 8
![Page 111: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/111.jpg)
3 27x
![Page 112: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/112.jpg)
3 27x
23 3 9x x x
![Page 113: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/113.jpg)
![Page 114: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/114.jpg)
ac bc2 2a b
3 3a b
3 3a b
GCF present
Difference of 2 SQUARES
Difference of 2 CUBES
Sum of 2 CUBES
![Page 115: GCF present Distributive Property in Reverse](https://reader036.vdocuments.us/reader036/viewer/2022062302/5a4d1ad87f8b9ab059973e9a/html5/thumbnails/115.jpg)
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