FOIL Method
When it comes to expanding a bracket like ( a + c )( x + y )
there is a simple way to remember all of the terms. This is the
word FOIL, and stands for
take products of the
First Outside Inside
Last
This is illustrated in the following.
Example
These terms are the products of the pairs highlighted below.
There are two other brackets that are worth remembering. These
are
( x + y)^{ 2}, which is a complete square,
and
( x + y)( x  y) , which is a difference of two
squares.
These are included in the following exercises.
Exercise
Remove the brackets from each of the following expressions
using FOIL.
(a) ( x + y)^{ 2}
(b) ( x + y)( x  y)
(c) ( x + 4)( x + 5)
(d) ( y + 1)( y + 3)
(e) (2 y + 1)( y  3)
(f) 2( x  3)^{ 2}  3( x + 1)^{ 2}
Solution
(a)
( x + y )^{
2 } 
= ( x + y )( x + y ) = x^{ 2}
+ xy + yx + y^{ 2} using FOIL
= x^{ 2} + 2xy + y^{ 2}

This is an IMPORTANT result and should be
committed to memory. Here x is the first member of the
the bracket and y is the second. The rule for the square
of ( x + y ), i.e. ( x + y )^{ 2} is
(b) Using FOIL again:
( x + y )( x 
y ) 
= x^{ 2}  xy + yx  y^{ 2} =
x^{ 2}  y^{ 2}

The form of the solution is the reason for the name difference
of two squares. This is another important result and is
worth committing to memory.
(c) Using FOIL :
( x + 4)( x +
5) 
= x^{ 2} + 5 x + 4 x + 20 = x^{
2} + 9 x + 20

(d) Using FOIL :
( y + 1)( y +
3) 
= y^{ 2} + 3 y + y + 3 = y^{
2} + 4 y + 3

(e) Using FOIL :
(2 y + 1)( y 
3) 
= 2 y^{ 2}  6 y + y  3 = 2 y^{
2}  5 y  3

(f) This one is best done in parts. First we
have
( x  3)^{ 2} = x^{ 2}  6 x + 9
and
( x + 1)^{ 2} = x^{ 2} + 2 x + 1
Thus
2( x  3)^{
2}  3( x + 1)^{ 2} 
= 2( x^{ 2}  6 x + 9)  3( x^{
2} + 2 x + 1) = 2 x^{ 2}  12 x + 18  3 x^{
2}  6 x  3
= 2 x^{ 2}  3 x^{ 2}  12 x  6 x +
18  3
=  x^{ 2}  18 x + 15

Quiz
To which of the following expressions does 9  ( x 3)^{ 2}
simplify?
(a)  x^{ 2}
(b) 6 x  x^{ 2}
(c) 18  x^{ 2}
(d) 6 x + x^{ 2}
Solution
First note that ( x  3)^{ 2} = x^{ 2}  6 x +
9, so
9  ( x  3)^{
2 } 
= 9  ( x^{ 2}  6 x + 9) = 9
 x^{ 2} + 6 x  9
= 9  9  x^{ 2} + 6 x
=  x^{ 2} + 6 x
= 6 x  x^{ 2}

