Using Patterns to Multiply Two Binomials
We obtain another useful pattern when we multiply the sum and difference
of the same two terms.
For example, letâ€™s use FOIL
to find
(a + b)(a  b). 
(a + b)(a  b) 
= 
F a^{2} 
 
O ab 
+ 
I ba 
 
L b^{2} 
When we combine like terms,
the middle terms cancel out. 

= a^{2}  b^{2} 
The expression a^{2}  b^{2} is called the difference of two squares
because the operation is subtraction and each term is a square.
The pattern tells us that the product of conjugates always results in the
difference of two perfect squares.
Formula â€”
The Product of the Sum and Difference of the Same Two Terms
Let a and b represent any real numbers.
(a + b)(a  b) = a^{2}  b^{2}
Note:
The following pairs of binomials are
examples of conjugates:
a + b and a  b
x  y and x + y
w + 2 and w  2
k  5 and k + 5
Example
Find: (m + 8n)(m  8n)
Solution
The expression (m + 8n)(m  8n) is in the form (a
+ b)(a  b). So we
can use the shortcut:

(a + b)(a  b) 
= a^{2}  b^{2} 
Substitute m for a and 8n for b. Simplify.
So, (m + 8n)(m  8n) = m^{2}  64n^{2}. 
(m + 8n)(m  8n) 
= (m)^{2}  (8n)^{2} = m^{2} 
64n^{2} 
