6.4
Special Factoring Techniques
Factor a difference of squares.
The formula for the product of the sum and difference of the
2
2
same two terms is  x  y  x  y   x  y .
Reversing this rule leads to the following special factoring rule.
x2  y 2   x  y  x  y 
For example, m2 16  m2  42   m  4 m  4  .
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).
EXAMPLE 1
Factoring Differences of
Squares
Factor each binomial if possible.
Solution:
t 2  81
  t  9 t  9
r 2  s2
  r  s  r  s 
16
x 
25
t 2  10
4 
4

  x   x  
5 
5

prime
t 2  36
prime
2
EXAMPLE 2
Factoring More Complex
Differences of Squares
Factor completely.
Solution:
50r 2  32
 2  25r 2  16 
 2  5r  4 5r  4 
49 x 2  25
64a 2  81b 2
z 4  81
  7 x  5 7 x  5
 8a  9b 8a  9b 
  z 2  9  z 2  9 
  z 2  9   z  3 z  3
Factor a perfect square trinomial.
The expressions 144, 4x2, and 81m6 are called perfect squares
2
2
2
6
3 2
because 144  12 , 4 x   2 x  , and 81m  9m .


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. We can multiply to
see that the square of a binomial gives one of the following
perfect square trinomials.
x 2  2 xy  y 2   x  y 
2
x  2 xy  y   x  y 
2
2
2
EXAMPLE 3
Factoring a Perfect Square
Trinomial
Factor k2 + 20k + 100.
Solution:
 k 2  20k  100
  k  10 
Check :
2  k 10  20k
2
Factoring Perfect Square Trinomials
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.
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.
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.
Factoring Perfect Square
Trinomials
EXAMPLE 4
Factor each trinomial.
Solution:
x  24 x  144
  x  12 
25 x  30 x  9
  5 x  3
36a 2  20a  25
prime
2
2
2
2
x  22x  121
2
9m  24m  16
2
Factor a difference of cubes.
Just as we factored the difference of squares in Objective 1,
we can also factor the difference of cubes by using the
following pattern.
positive
x3  y 3   x  y   x 2  xy  y 2 
same sign
opposite sign
This pattern for factoring a difference of cubes should be memorized.

EXAMPLE 5
Factoring Differences of Cubes
positive
Factor each polynomial.
x3  y 3   x  y   x 2  xy  y 2 
same sign
x3 – y 3
x 3  216  x 3  6 3
x–y
opposite sign
x2 + xy + y2
  x  6   x 2  6 x  36 
x3 – y 3
4m3  32  4m 3  8 4m3  23 
x–y
x2 + xy + y2
 4m  2m 2  2m  4
positive
x3  y 3   x  y   x 2  xy  y 2 
same sign
p3  64
27 x3  64 y 3
opposite sign
  p  4   p 2  4 p  16 
  3 x  4 y   9 x 2  12 xy  16 y 2 
The methods of factoring discussed in this section are
summarized here.