Difference of Two Squares (DOTS):
If we were to use FOIL on the following:
We would result in the following:
(a + b)(a – b)
a 2  ab  ab  b 2
a2  b2
Thus when we factor this difference of two squares, we will finish with the same product
(a + b)(a – b).
m 2  16
Ex:
m2  42
(m  4)( m  4)
Let’s think back to factoring polynomials. What does the subtraction sign (-) before the
last term tell us about the signs in our two binomials?
( + )( - )
After we have set up the signs, we can then place the square root of each term in both
binomials. (Remember: the square root of a number undoes squaring a number)
(m + 4)(m – 4)
IMPORTANT! ~~ In order for these special factoring cases to be consider DOTS, both
terms must be squares, and a subtraction sign must live between them!
x 2  9 GOOD! =)
Ex:
x 2  49
y 2  m2
9
z2 
16
2
x 8
p 2  16
x 2  16 SOTS! =(
x 2  8 BAD! =(
p 2  100
25
x2 
36
2
x  y2
9m 2  49
64a 2  25
25m 2  16
49 z  64
2
81y  36
2
9x  4z
2
p  36
4
m  16
4
2
50r 2  32
27 y 2  75
25a 2  64b 2
k 4  49
81r 4  16
Perfect Square Trinomials:
Q: Why do we call numbers such as 144, 4 x 2 , and 81m 2 perfect squares?
144  12 2
4 x 2  (2 x) 2
81m 6  (9m 3 ) 2
Def: A perfect square trinomial is the square of a binomial.
x 2  8 x  16
Ex: ( x  4)( x  4)
( x  4) 2
IMPORTANT ~~ In order for a trinomial to be a perfect square trinomial, two of its
terms must be perfect squares. HOWEVER, this does not guarantee the polynomial will
be a perfect square.
16 x 2  4 x  15
NO GOOD! Only 1 perfect square term.
Ex:
x 2  10 x  25
p 2  18 p  81
x 2  22 x  121
16a 2  56a  49
p 2  14 p  49
121 p 2  110 p  100
m 2  8m  16
64 x 2  48 x  9
x 2  2x  1
27 y 3  72 y 2  48 y