## Using the FOIL Technique to Multiply Binomials

Read this article, which gives many examples of using the FOIL technique to multiply two binomials. Then, try some practice problems.

Recall the the distributive law: for all real numbers $a$, $b$, and $c$, $a(b+c)=ab+ac$.

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 is how treat it as a singleton 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 are taking something that seems to have two parts, and you are treating it as a single thing, a singleton!

Look what happens:

 $(a+b)(c+d)$ $=\overset{z}{\overbrace{(a+b)}}(c+d)$ Give $(a+b)$ the name $z$ $=z(c+d)$ Rewrite $=zc+zd$ Use the distributive law $=(a+b)c+(a+b)d$ Since $z=a+b$ $=ac+bc+ad+bd$ Use the distributive law twice $=ac+ad+bc+bd$ Re-order; switch the two middle terms $=\underset{F}{\underbrace{ac}}+\underset{O}{\underbrace{ad}}+\underset{I}{\underbrace{bc}}+\underset{L}{\underbrace{bd}}$

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 is 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 expression $(a+b)(c+d)$ 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 expression $(a+b)(c+d)$ 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:

$(x−1)(x+4)$

$=\underset{F}{\underbrace{(x\cdot x)}}+\underset{O}{\underbrace{(x\cdot 4)}}+\underset{I}{\underbrace{(-1\cdot x)}}+\underset{L}{\underbrace{(-1\cdot 4)}}$

$=x^{2}+4x-x-4$

$=x^{2}+3x-4$

You want to be able to write this down without including the first step above:

$=(x-1)(x+4)=\underset{F}{\underbrace{x^{2}}}+\underset{O}{\underbrace{4x}}-\underset{I}{\underbrace{x}}-\underset{L}{\underbrace{4}}=x^{2}+3x-4$

Then, after you have practiced a bit, you want to be able to combine the ‘outers’ and ‘inners’ in your head,

and write it down using only one step:

$(x-1)(x+4)=\underset{F}{\underbrace{x^{2}}}+\underset{OI}{\underbrace{3x}}-\underset{L}{\underbrace{4}}$

### Examples

Simplify: $(x+3)(x-2)$

Answer: $x^{2}+x-6$

Simplify: $(x+4)(x-4)$

Answer: $x{^2}-16$

#### Practice Questions

Answers must be written in the most conventional way:
$x^{2}$ term first, $x$ term next, constant term last.

For example, type 'x^2' for $x^2$.

Simplify the equation below:

Source: Tree of Math, https://www.onemathematicalcat.org/algebra_book/online_problems/foil_1x.htm