3.3 special factoring 12/5/2012. perfect squares 11 1 42 2 93 3 164 4 255 5 366 6 49 7 7 648 8 819 9...
TRANSCRIPT
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