GCF present Distributive Property in Reverse.

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

2 11m m

11m m


Example 1

m

2 11m m

11m m


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

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

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

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 .


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 .


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

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

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

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

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

327 1x

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

22 2 4x x x

3 8x 22 2 4x x x

x2 –2x 4

x

2

3x 22x 4x22x 4x 8

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

GCF present Distributive Property in Reverse.

Documents

Transcript of GCF present Distributive Property in Reverse.