Download Factoring Perfect Square Trinomials a

6.4
1.
2.
3.
4.
Factoring Special Products
Factor perfect square trinomials.
Factor a difference of squares.
Factor a difference of cubes.
Factor a sum of cubes.
Write as many perfect squares as you can.
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
625
Write as many perfect cubes as you can.
1
8
27
64
125
Perfect Square Trinomials:
Perfect squares
x  3x  3 
x  6x  9
2
x  3
2
6x is double the product.
Perfect squares
2 x  3 y 2 x  3 y   4 x
2
 12 xy  9 y
2
2 x  3 y 
2
-12xy is double the product.
Perfect Square Trinomials:
Caution: Don’t just check the first and last terms!
x  3x  12 
x  15x  36
2
 x  6
2
15x is not double the
product.
Factor completely :
Perfect squares
4a  20ab  25b
2
2
2a  5b 
2
-20ab is double the
product.
Check by foiling!
2a  5b2a  5b  4a
2
 20ab  25b
2
Factor completely :
Perfect squares
2
64a  208a  169
8a  132
-208a is double the
product.
Check by foiling!
Factor completely :
9m  24m  16
2
3m  4
2
24m is double the
product.
Check by foiling!
3m  4
2
 9m  24m  16
2
Factor completely :
y  6 y  36
2
 y  6
2
6 is NOT double the product.
Not a perfect square trinomial.
It may still be factorable.
Prime
Factor completely :
54 x  72 x  24
2

6 9 x  12 x  4
2
63x  2
2

Factoring Perfect Square Trinomials
a2 + 2ab + b2 = (a + b)2
a2 – 2ab + b2 = (a – b)2
Difference of Squares:
x  3x  3 
Conjugates
x 9
2
 x  3 x  3
2 x  3 y 2 x  3 y   4 x
Conjugates
2
 9y
2
2 x  3 y 2 x  3 y 
Factor completely :
a  121
2
Think Conjugates!
a  11a  11
Check by foiling!
Factor:
25x  16
2
Think Conjugates
5 x  45 x  4
Check by foiling!
Factor completely :
16 x  49
2
The sum of squares CANNOT be factored!
Prime
Factor completely :
64m  36 y
2
2
8m  6 y 8m  6 y 

4 16m  9 y
2
2

44m  3 y 4m  3 y 
Check by foiling!
Factor completely:
x  16
4
x
2

4 x 4
2
x  2x  2x
2

4
Check by foiling!

Factoring a Difference of Squares
a2 – b2 = (a + b)(a – b)
Warning: A sum of squares a2 + b2 is
prime and cannot be factored.
Copyright © 2011 Pearson Education, Inc.
Sum and Difference of Cubes
Same.
Multiply:
x y 
3
3
Cube Root
x y 
3
3
3 terms – trinomial rather
Opposite.than binomial
x  y x
x  y x
Always positive
2
Square
2
 xy  y

2
 xy  y
2
Product Square


Factor completely:
8a  27b
3
3
Cubes = trinomial
 2a  3b 4a
2
Square
 6ab  9b

Product
2
Square

Factor completely:
y  64
3
Cubes = trinomial

y 4
 y
2
Square
 4 y  16 

Product
Square
Factor completely:
1000a  27b
3
3
Cubes = trinomial
10a  3b100a  30ab  9b 
2
Square
2

Product
Square
Factoring a Sum or Difference of Cubes
a3 + b3 = (a + b)(a2 – ab + b2)
a3 – b3 = (a – b)(a2 + ab + b2)
Copyright © 2011 Pearson Education, Inc.
Factor completely. 9x2 – 49
a) (3x + 5)2
b) (3x + 7)(3x – 7)
c) (3x – 7)2
d) (7x + 3)(7x – 3)
6.4
Copyright © 2011 Pearson Education, Inc.
Slide 6- 23
Factor completely. 9x2 – 49
a) (3x + 5)2
b) (3x + 7)(3x – 7)
c) (3x – 7)2
d) (7x + 3)(7x – 3)
6.4
Copyright © 2011 Pearson Education, Inc.
Slide 6- 24
Factor completely. 4a2 – 20a + 25
a) (2a + 5)2
b) (2a – 5)2
c) (4a + 5)2
d) (4a – 5)2
6.4
Copyright © 2011 Pearson Education, Inc.
Slide 6- 25
Factor completely. 4a2 – 20a + 25
a) (2a + 5)2
b) (2a – 5)2
c) (4a + 5)2
d) (4a – 5)2
6.4
Copyright © 2011 Pearson Education, Inc.
Slide 6- 26
Factor completely. 2n2 + 24n + 72
a) 2(n + 6)2
b) 2(n + 6)(n – 6)
c) 2(n – 6)2
d) (2n + 6)(2n – 6)
6.4
Copyright © 2011 Pearson Education, Inc.
Slide 6- 27
Factor completely. 2n2 + 24n + 72
a) 2(n + 6)2
b) 2(n + 6)(n – 6)
c) 2(n – 6)2
d) (2n + 6)(2n – 6)
6.4
Copyright © 2011 Pearson Education, Inc.
Slide 6- 28