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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Last modified: Wednesday, May 5, 2021, 4:55 PM