3.3SPECIAL Factoring

12/5/2012

Perfect Squares

11

1

42

2

93

3

164

4

255

5

366

6

497

7

648

8

819

9

10010

10

12111

11

14412

12

16913

13

Review

Find the product using FOIL1.(x + 2) (x – 2) Answer: x2 – 42. (x + 5) (x – 5)Answer: x2 – 253. (2x – 3) (2x + 3)Answer: 4x2 – 9What’s the pattern???

Difference of Two Squares Pattern

(a + b) (a – b) = a2 – b2

In reverse, a2 – b2 gives you (a + b) (a – b)Examples: 1. x2 – 4 = x2 – 22 = (x + 2) (x – 2)

2. x2 – 144 =(x + 12) (x – 12)3. 4x2 – 25 = (2x + 5) (2x – 5)

1000 = 103

729 = 93

512 = 83

343 = 73

216 = 63

Perfect Cubes125 = 53

64 = 43

27 = 33

8 = 23

1 = 13

2233 babababa

2233 babababa

The sum of two cubes:

The difference of two cubes:

Factor the Sum or Difference of Two Cubes

a. Factor .x 3 + 64 b. Factor .8p 3 – q 3

SOLUTION

Write as sum of two cubes.

x 3 + 64 = x 3 + 43a.

( )4x + ( )x 2 4x +– 42= Use special product pattern.

( )4x + ( )x 2 4x +– 16= Simplify.

2233 babababa

Factor the Sum or Difference of Two Cubes

= –( )q2p + q22pq+4p2( ) Simplify.

b. 8p 3 – q 3 –( )2p 3 q 3= Write as difference of two cubes.

= –( )q2p + q22pq[ ]( )2p 2 + Use special product pattern.

Checkpoint Factor the polynomial.

1. x 3 + 1

2. 125x 3 + 8

ANSWER

( )1x + ( )x 2 x +– 1

( )25x + ( )25x 2 10x +– 4

3. x 3 216– ( )6x +( )x 2 6x + 36–

2233 babababa 2233 babababa

Factor Polynomials with GCF

a. Factor 16x 4 2x.–

Take out GCF.= ( )2x 8x 3 1–a. 16x 4 2x–

Use a3 –b3 pattern.= ( )2x 2x 1– 4x 2 2x 1+ +( )

Factor by Grouping

Factor the polynomial.

b. a. x 2 ( )1x – ( )1x –9– x 3 2x 2 16x– – 32+

SOLUTION

Use distributive property.

a. x 2 ( )1x – ( )1x –9– = ( )9x 2 – ( )1x –

Difference of two squares

= ( )3x – ( )3x + ( )1x –

Factor by Grouping

Factor each group.

= )x 2 – ( –2 + 16( x ) )– 2( x

Use distributive property.

= )– 16( )– 2( xx 2

Difference of two squares

= ( )4x – ( )4x + ( )2x –

Group terms.

b.

= ( )x 3 – ( )32–x 3 2x 2 16x– – 32+ 2x 2 + 16x +

Checkpoint

Factor the polynomial by grouping.

8.

Factor by Grouping

x 2 ( )6x + ( )6x +4–

9. x 3 4x 2 25x– – 100+

10.

x 3 3x 2 4x 12++ +

ANSWERS

( )2x – ( )2x +( )6x +

( )5x – ( )5x +( )4x –

( )3x + ( )4x 2 +

Homework:

Worksheet 3.3