factoring

1. Chapter 6 Section 4

2. Objectives
1
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Special Factoring Techniques
Factor a difference of squares.
Factor a perfect square trinomial.
Factor a difference of cubes.
Factor a sum of cubes.
6.4
2
3
4

3. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
By reversing the rules for multiplication of binomials from Section
5.6, we get rules for factoring polynomials in certain forms.
Special Factoring Techniques
Slide 6.4-3

4. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 1
Factor a difference of squares.
Slide 6.4-4

5. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Factor a difference of squares.
The formula for the product of the sum and difference of the same two
terms is
Factoring a Difference of Squares
   2 2
.
x y x y x y
   
  
2 2
x y x y x y
   
The following conditions must be true for a binomial to be a difference
of squares:
1. Both terms of the binomial must be squares, such as
x2
, 9y2
, 25, 1, m4
.
2. The second terms of the binomials must have different signs (one
positive and one negative).
  
2 2 2
16 4 4 4 .
m m m m
     
For example,
Slide 6.4-5

6. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Factor each binomial if possible.
Solution:
2
81
t 
2 2
r s

2
10
y 
2
36
q 
  
9 9
t t
  
  
r s r s
  
prime
prime
After any common factor is removed, a sum of squares cannot
be factored.
Slide 6.4-6
EXAMPLE 1 Factoring Differences of Squares

7. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Factor each difference of squares.
Solution:
2
49 25
x 
2 2
64 81
a b

  
7 5 7 5
x x
  
  
8 9 8 9
a b a b
  
You should always check a factored form by multiplying.
Slide 6.4-7
EXAMPLE 2 Factoring Differences of Squares

8. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Factor completely.
Solution:
2
50 32
r 
4
100
z 
4
81
z 
 
2
2 25 16
r
 
  
2 2
10 10
z z
  
  
2 2
9 9
z z
  
  
2 5 4 5 4
r r
  
Factor again when any of the factors is a difference of squares as in
the last problem.
Check by multiplying.
   
2
9 3 3
z z z
   
Slide 6.4-8
EXAMPLE 3 Factoring More Complex Differences of Squares

9. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 2
Factor a perfect square trinomial.
Slide 6.4-9

10. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
The expressions 144, 4x2
, and 81m6
are called perfect squares
because
A perfect square trinomial is a trinomial that is the square of a
binomial. A necessary condition for a trinomial to be a perfect square
is that two of its terms be perfect squares.
Even if two of the terms are perfect squares, the trinomial may not be
a perfect square trinomial.
Factor a perfect square trinomial.
Factoring Perfect Square Trinomials
 
2
2 2
2
x xy y x y
   
 
2
2 2
2
x xy y x y
   
2
144 ,
12
  
2
2
4 2 ,
x x
  
2
6 3
81 9 .
m m

and
Slide 6.4-10

11. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Solution:
 
2
10
k
 
Factor k2
+ 20k + 100.
2
20 100
k k
  
Check :
2 10 20
k k
  
Slide 6.4-11
EXAMPLE 4 Factoring a Perfect Square Trinomial

12. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Factor each trinomial.
Solution:
2
24 144
x x
 
2
25 30 9
x x
 
2
36 20 25
a a
 
 
2
12
x
 
 
2
5 3
x
 
prime
3 2
18 84 98
x x x
   
2
2 9 42 49
x x x
    
2
2 3 7
x x
 
Slide 6.4-12
EXAMPLE 5 Factoring Perfect Square Trinomials

13. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
1. The sign of the second term in the squared binomial is always the
same as the sign of the middle term in the trinomial.
Factoring Perfect Square Trinomials
3. Perfect square trinomials can also be factored by using grouping or
the FOIL method, although using the method of this section is often
easier.
2. The first and last terms of a perfect square trinomial must be
positive, because they are squares. For example, the polynomial
x2
– 2x – 1 cannot be a perfect square, because the last term is
negative.
Slide 6.4-13

14. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 3
Factor a difference of cubes.
Slide 6.4-14

15. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Factor a difference of cubes.
The polynomial x3
− y3
is not equivalent to (x − y )3
,
whereas
     
3
x y x y x y x y
    
  
2 2
2
x y x xy y
   
  
3 3 2 2
x y x y x xy y
    
Factoring a Difference of Cubes
  
3 3 2 2
x y x y x xy y
    
same sign
positive
opposite sign
Slide 6.4-15
This pattern for factoring a difference of cubes should be
memorized.

16. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Factor each polynomial.
Solution:
3
216
x 
3
27 8
x 
3
5 5
x 
  
2
6 6 36
x x x
   
  
2
3 2 9 6 4
x x x
   
3 6
64 125
x y
   
2 2 2 4
4 5 16 20 25
   
x y x xy y
 
3
5 1
x
    
2
5 1 1
x x x
   
A common error in factoring a difference of cubes, such as
x3
− y3
= (x − y)(x2
+ xy + y2
), is to try to factor x2
+ xy + y2
. It is easy to
confuse this factor with the perfect square trinomial x2
+ 2xy + y2
. But because
there is no 2, it is unusual to be able to further factor an expression of the
form x2
+ xy +y2
.
Slide 6.4-16
EXAMPLE 6 Factoring Differences of Cubes

17. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objective 4
Factor a sum of cubes.
Slide 6.4-17

18. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Factor a sum of cubes.
A sum of squares, such as m2
+ 25, cannot be factored by using real
numbers, but a sum of cubes can.
Note the similarities in the procedures for factoring a sum of cubes and a
difference of cubes.
1. Both are the product of a binomial and a trinomial.
2. The binomial factor is found by remembering the “cube root, same sign,
cube root.”
3. The trinomial factor is found by considering the binomial factor and
remembering, “square first term, opposite of the product, square last
term.”
Factoring a Sum of Cubes
  
3 3 2 2
x y x y x xy y
    
same sign
positive
opposite sign
Slide 6.4-18

19. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Methods of factoring discussed in this section.
Slide 6.4-19

20. Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Factor each polynomial.
Solution:
3
64
p 
3 3
27 64
x y

6 3
512a b

  
2
4 4 16
p p p
   
  
2 2
3 4 9 12 16
x y x xy y
   
  
2 4 2 2
8 64 8
a b a a b b
   
Slide 6.4-20
EXAMPLE 7 Factoring Sums of Cubes