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BASIC FOIL
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The concepts for this exercise are summarized below.
For a complete discussion, read the text.
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At first glance, it might not look like the distributive law applies
to the expression (a+b)
(c+d) .
However, it does—once you apply a popular mathematical technique called "treat it as a singleton".
Here's how it goes:
First, rewrite the distributive law using some different variable names:
z(c+d
)=zc+zd .
This says that anything times (c+d)
is the anything times c ,
plus the anything times d .
Now, look back at (a+b)
(c+d) ,
and take the group (a+b)
as z .
That is, you're taking something that seems to have two parts,
and you're treating it as a single thing, a "singleton"!
Look what happens:
(a+b)
(c+d)
|
=
(a+b)
⏞
"z"
(c+d) |
(give a+b the name
z ) |
|
=z(c+
d)
|
(rewrite) |
|
=zc+zd
|
(use the distributive law) |
|
=(a+b
)c+(a+
b)d
|
(z=a+b
) |
| =ac+bc+
ad+bd |
(use the distributive law twice) |
| =ac+ad+
bc+bd |
(re-order) |
|
=
ac
⏟
First
+
ad
⏟
Outer
+
bc
⏟
Inner
+
bd
⏟
Last
|
|
You get four terms, and each of these terms is assigned a letter.
These letters form the word FOIL ,
and provide a powerful memory device
for multiplying out expressions of the form
(a+b)
(c+d) .
Here's the meaning of each letter in the word FOIL:
-
The first number in the group
(a+b)
is
a ;
The first number in the group
(c+d)
is
c .
Multiplying these Firsts together gives
ac ,
which is labeled " F ".
-
When you look at the group
(a+b)
(c+d)
from far away,
you see a and
d on the outside.
That is, a and d
are the outer numbers.
Multiplying these Outers together gives
ad ,
which is labeled " O ".
-
Similarly, when you look at the group
(a+b)
(c+d)
from far away,
you see b and
c on the inside.
That is, b and c
are the inner numbers.
Multiplying these Inners together gives
bc ,
which is labeled " I ".
-
The last number in the group
(a+b)
is
b ;
The last number in the group
(c+d)
is
d .
Multiplying these Lasts together gives
bd ,
which is labeled " L ".
One common application of FOIL is to multiply out expressions
like
(x-1)
(x+4) .
Remember the exponent laws, and be sure to combine like terms whenever possible:
(x-1)
(x+4)=
x2
+4x-x-
4=x2
+3x-4
EXAMPLES:
Simplify: (x+3)(x-2) Answer: x^2 + x - 6
Simplify: (x+4)(x-4) Answer: x^2 - 16