Factoring the Sum and Difference of Two cubes

a3 + b3

a3 – b3

Count

• How long is the edge?

• How many squares in the face?

• How many blocks?

1

1

1

Count

Edge Face Blocks

1 1 12 4 8

Count

Edge Face Blocks

1 1 12 4 83 9 27

Count

Edge Face Blocks

1 1 12 4 83 9 274 16 645 25 125

n n2 n3

Memorize the First 10 Perfect Cubesn n2 n3

1 1 12 4 83 9 274 16 645 25 1256 36 2167 49 3438 64 5129 81 729

10 100 1000

Recall the Difference of Two Squares Formula

a2 – b2

=(a + b)(a – b)x2 – 9 =(x + 3)(x – 3)

• There are similar formulas for the sum and difference of two cubes.

Multiply a Binomial by a Trinomial2 2( )x xy y

( )x y3x 2x y 2xy

2x y 2xy 3y3x 3y

3 3x y2 2( )( )x y x xy y

The Sum of Cubes

Difference of Cubes2 2( )x xy y

( )x y3x 2x y 2xy

2x y 2xy 3y3x 3y

3 3x y2 2( )( )x y x xy y

Compare the Formulas

3 3x y 2 2( )( )x y x xy y

The Sum of Cubes

3 3x y 2 2( )( )x y x xy y

The Difference of Cubes

They are just alike except for where they are different.

Using the Difference of Cubes

x3 - 8Recall 23 = 8

= (x - 2)(x2 + 2x + 4)

3 3x y 2 2( )( )x y x xy y

Using the Sum of Cubes

y3 + 27Recall 33 = 27

= (y + 3)(y2 – 3y + 9)

3 3x y 2 2( )( )x y x xy y

Factor Out the Common Factor

x(3a + 2) + 7(3a + 2) = (3a + 2)(x+7)

3xa + 2x + 21a + 14 =

3xa + 2x + 3(7)a + 2(7) =

This is called factoring by grouping.

What is factoring by grouping?Factoring a common monomial from pairs of terms, then looking for a common binomial factor is called factor by grouping.

When do I use factoring by grouping?*when the problem consists of 4 terms

How will my answer look?*it will be the product of two binomials

Factor the expression

25 ( 2) 3( 2)x x x

25 ( 2) 3( 2)x x x

2x

Notice there are two terms

Notice what each term has

in common.Pull the common factor out of each term.

( 2)x Notice what is left in each term after factoring out the common factor.

2(5 3)x

Try this example:

7 ( 5) 3( 5)y y y

( 5)(7 3)y y

Factor the polynomial3 27 2 14m m m 3 2( 7 ) ( 2 14)m m m

Form two binomials with a + sign between them.

2( 7 2) ( 7)mm m 2( 2( 7) )mm

Try this example:

29 ( 1) 7( 1)x x x

3 29 9 7 7x x x 3 2(9 9 ) ( 7 7)x x x

2( 1)(9 7)x x

6x2 – 3x – 4x + 2 by grouping

6x2 – 3x – 4x + 2= (6x2 – 3x) + (– 4x + 2)= 3x(2x – 1) + -2(2x - 1)

= (2x – 1)(3x – 2)

HomeworkWB pp 89 and 90

Book p. 78 #1-27 0dd, p. 79 #1-27 odd

Page 78

Page 78