## 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 , , and , .

At first glance, it might not look like the distributive law applies to the expression .

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: .

This says that **anything** times is the **anything** times , plus the **anything** times .

Now, look back at , and take the group as .

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:

Give the name | |

Rewrite | |

Use the distributive law | |

Since | |

Use the distributive law twice | |

Re-order; switch the two middle terms | |

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 .

Here is the meaning of each letter in the word **FOIL**:

- The first number in the group is ;

the first number in the group is .

Multiplying these**Firsts**together gives , which is labeled . - When you look at the expression from far away,

you see and on the**outside**.

That is, and are the**outer**numbers.

Multiplying these**Outers**together gives , which is labeled . - Similarly, when you look at the expression from far away,

you see and on the**inside**.

That is, and are the**inner**numbers.

Multiplying these**Inners**together gives , which is labeled . - The last number in the group is ;

the last number in the group is .

Multiplying these**Lasts**together gives , which is labeled .

One common application of FOIL is to multiply out expressions like .

Remember the exponent laws, and be sure to combine like terms whenever possible:

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

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:

Examples

Write your answer in the most conventional way.

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

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