5.4 Special
Factoring
Techniques
Special Factoring Techniques
By reversing the rules for multiplication of binomials from Section 4.6, we
get rules for factoring polynomials in certain forms.
Slide 5.4-3
Objective 1
Factor a difference of squares.
Slide 5.4-4
Factor a difference of squares.
The formula for the product of the sum and difference of the same two
2
2
terms is
 x  y  x  y   x
y .
Factoring a Difference of Squares
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).
Slide 5.4-5
CLASSROOM
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 
y 2  10
prime
q 2  36
prime
After any common factor is removed, a sum of squares cannot
be factored.
Slide 5.4-6
CLASSROOM
EXAMPLE 2
Factoring Differences of Squares
Factor each difference of squares.
Solution:
49 x 2  25
  7 x  5 7 x  5
64a 2  81b 2
 8a  9b 8a  9b 
You should always check a factored form by multiplying.
Slide 5.4-7
CLASSROOM
EXAMPLE 3
Factoring More Complex Differences of Squares
Factor completely.
Solution:
50r  32
 2  25r 2  16 
z 4  100
  z 2  10  z 2  10 
z 4  81
  z  9  z  9 
2
2
2
 2  5r  4 5r  4 
  z 2  9   z  3 z  3
Factor again when any of the factors is a difference of squares as in the last
problem.
Check by multiplying.
Slide 5.4-8
Objective 2
Factor a perfect square trinomial.
Slide 5.4-9
Factor a perfect square trinomial.
The expressions 144, 4x2, and 81m6 are called perfect squares because
144  12 , 4 x   2 x  ,
2


3 2
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.
2
2
6
Even if two of the terms are perfect squares, the trinomial may not be a
perfect square trinomial.
Factoring Perfect Square Trinomials
2
2
2
x  2 xy  y   x  y 
x 2  2 xy  y 2   x  y 
2
Slide 5.4-10
CLASSROOM
EXAMPLE 4
Factoring a Perfect Square Trinomial
Factor k2 + 20k + 100.
Solution:
 k 2  20k  100
  k  10 
2
Check :
2  k 10  20k
Slide 5.4-11
CLASSROOM
EXAMPLE 5
Factoring Perfect Square Trinomials
Factor each trinomial.
Solution:
x 2  24 x  144
  x  12 
2
25 x 2  30 x  9
  5 x  3
2
36a 2  20a  25
prime
18 x  84 x  98 x
 2 x  9 x 2  42 x  49 
3
2
 2 x  3x  7 
2
Slide 5.4-12
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.
Slide 5.4-13
Objective 3
Factor a difference of cubes.
Slide 5.4-14
Factor a difference of cubes.
Factoring a Difference of Cubes
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.
The polynomial x3 − y3 is not equivalent to (x − y )3,
 x  y
whereas
3
  x  y  x  y  x  y 
  x  y   x 2  2 xy  y 2 
x3  y 3   x  y   x 2  xy  y 2 
Slide 5.4-15
CLASSROOM
EXAMPLE 6
Factoring Differences of Cubes
Factor each polynomial.
Solution:
x3  216
  x  6   x 2  6 x  36 
27 x  8
  3x  2   9 x 2  6 x  4 
5x  5
 5  x 3  1
3
3
64 x  125 y
3
6
 5  x  1  x 2  x  1
  4 x  5 y 2 16 x 2  20 xy 2  25 y 4 
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 5.4-16
Objective 4
Factor a sum of cubes.
Slide 5.4-17
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.
Factoring a Sum of Cubes
positive
x3  y 3   x  y   x 2  xy  y 2 
same sign
opposite sign
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.”
Slide 5.4-18
Methods of factoring discussed in this section.
Slide 5.4-19
CLASSROOM
EXAMPLE 7
Factoring Sums of Cubes
Factor each polynomial.
Solution:
  p  4   p 2  4 p  16 
p3  64
27 x  64 y
3
512a  b
6
3
3
  3 x  4 y   9 x  12 xy  16 y
2
2

  8a 2  b  64a 4  8a 2b  b 2 
Slide 5.4-20