§ 5.5
Factoring Special Forms
The Difference of Two Squares
The Difference of Two Squares
If A and B are real numbers, variables, or algebraic
expressions, then
A2  B2   A  B A  B .
In words: The difference of the squares of two terms,
factors as the product of a sum and a difference of those
terms.
Blitzer, Intermediate Algebra, 4e – Slide #67
The Difference of Two Squares
EXAMPLE
Factor: 25x 4  9 y 6.
SOLUTION
We must express each term as the square of some monomial.
Then we use the formula for factoring A2  B2   A  B A  B.
25x 4  9 y 6
5x   3 y 
2 2
5x
2
3 2

 3 y 3 5x 2  3 y 3
Express as the difference of
two squares

Factor using the Difference
of Two Squares method
Blitzer, Intermediate Algebra, 4e – Slide #68
The Difference of Two Squares
EXAMPLE
Factor: 6 x 2  6 y 2.
SOLUTION
The GCF of the two terms of the polynomial is 6. We begin by
factoring out 6.
6x2  6 y 2

6 x2  y 2

Factor the GCF out of both
terms
6x  y x  y 
Factor using the Difference
of Two Squares method
Blitzer, Intermediate Algebra, 4e – Slide #69
The Difference of Two Squares
EXAMPLE
Factor completely: x 4  1.
SOLUTION
  1
x 1  x
4
2 2
2




Express as the difference of
two squares

 x 2  1 x2 1
 x 2  1 x2 12

The factors are the sum and
difference of the expressions
being squared
The factor x 2  1 is the
difference of two squares and
can be factored
Blitzer, Intermediate Algebra, 4e – Slide #70
The Difference of Two Squares
CONTINUED


 x 2  1 x  1x 1

The factors of x 2  1 are the
sum and difference of the
expressions being squared

Thus, x 4 1  x 2  1 x  1x 1 .
Blitzer, Intermediate Algebra, 4e – Slide #71
Factoring Completely
EXAMPLE
Factor completely: x3  3x 2  9 x  27.
SOLUTION
x 3  3x 2  9 x  27


 x3  3x 2   9x  27
 x 2 x  3  9x  3

 x  3 x 2  9

Group terms with common
factors
Factor out the common factor
from each group
Factor out x + 3 from both
terms
 x  3x  3x  3
Factor x 2  9 as the difference
of two squares
Blitzer, Intermediate Algebra, 4e – Slide #72
Factoring Special Forms
Factoring Perfect Square Trinomials
Let A and B be real numbers, variables, or algebraic
expressions.
1) A2  2 AB  B2   A  B
2
2) A2  2 AB  B2   A  B
2
Blitzer, Intermediate Algebra, 4e – Slide #73
Factoring Perfect Square Trinomials
EXAMPLE
Factor: 16x 2  40xy  25y 2.
SOLUTION
We suspect that 16x 2  40xy  25y 2 is a perfect square trinomial
because 16x2  4x2 and 25y 2   5 y 2. The middle term can
be expressed as twice the product of 4x and -5y.
16x 2  40xy  25y 2
 4x  2  4x  5 y    5 y 
Express in A2  2 AB  B 2 form
 4x  5 y 
Factor
2
2
2
Blitzer, Intermediate Algebra, 4e – Slide #74
Grouping & Difference of Two Squares
EXAMPLE
Factor: x 4  x 2  6 x  9.
SOLUTION
x4  x2  6x  9


 x4  x2  6x  9
 x 4  x  3
2
   x  3
 x
2 2
2
Group as x 4 minus a perfect
square trinomial to obtain a
difference of two squares
Factor the perfect square
trinomial
Rewrite as the difference of
two squares
Blitzer, Intermediate Algebra, 4e – Slide #75
Grouping & Difference of Two Squares
CONTINUED



 x 2  x  3 x 2  x  3



 x2  x  3 x2  x  3
Factor the difference of two
squares. The factors are the
sum and difference of the
expressions being squared.
Simplify
Thus, x4  x2  6x  9  x2  x  3x2  x  3.
Blitzer, Intermediate Algebra, 4e – Slide #76
The Sum & Difference of Two Cubes
Factoring the Sum & Difference of Two Cubes
1)
Factoring the Sum of Two Cubes:

A3  B3   A  B A2  AB  B2
Same Signs
Opposite Signs
2) Factoring the Difference of Two Cubes:

A3  B3   A  B A2  AB  B2
Same Signs

Opposite Signs
Blitzer, Intermediate Algebra, 4e – Slide #77

The Sum & Difference of Two Cubes
EXAMPLE
Factor: x3 y 3  64.
SOLUTION
x3 y3  64
3
 xy  43

 xy  4 xy  xy4  42

2

 xy  4 x 2 y 2  4xy  16

Rewrite as the Sum of Two
Cubes
Factor the Sum of Two Cubes
Simplify
Thus, x3 y3  64  xy  4x 2 y 2  4xy  16.
Blitzer, Intermediate Algebra, 4e – Slide #78
The Sum & Difference of Two Cubes
EXAMPLE
Factor: 125x6  64y 6.
SOLUTION
125x6  64y 6
   4 y 
 5x

2 3
2 3
Rewrite as the Difference of
Two Cubes
        Factor the Difference of Two
2
 5x 2  4 y 2 5x 2  5x 2 4 y 2  4 y 2


 5x 2  4 y 2 25x 4  20x 2 y 2  16y 4

2
Cubes
Simplify
Thus, 125x6  64y 6  5x2  4 y 2 25x4  20x2 y 2  16y 4 .
Blitzer, Intermediate Algebra, 4e – Slide #79