A binomial is a polynomial with two terms that are either added or subtracted from each other like \(3x+2\) or even something more complicated like \(5xy-3x^2\). When multiplying any two polynomials, the ultimate goal is to make sure every term in the first polynomial has been multiplied by every term in the second polynomial. When multiplying binomials, this process can be simplified into an easy to remember process that you have probably heard of: FOIL. In this lesson, we will review this process and look at some examples of applying it.

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## What does FOIL stand for?

FOIL is shorthand for remembering the process of multiplying binomials. The letters stand for:

**F**irst: multiply the first terms of each binomial

**O**uter: multiply the “outer” terms in each binomial

**I**nner: multiply the “inner” terms in each binomial

**L**ast: multiply the “last” terms in each binomial

If you think about it, this makes sure every term in each binomial is multiplied by every other term. Take the first term for instance. Once you have FOILed, it has been multiplied by the first term of the second binomial and the last (by the “outer” step). This is true for every term!

Once this process is complete, it is very likely you have some cleaning up to do in the sense of combining like terms, so you should make sure to simplify your answer once you are done.

## Examples of multiplying binomials using FOIL

In the following examples, we will apply this method to multiply some binomials. Pay close attention to how we apply each part of FOIL and also how we simplify to get the final answer.

### Example

Multiply: \((3x+1)(2x-3)\)

### Solution

For this first example, we will look at every step individually.

**First:** The first two terms are the \(3x\) and the \(2x\). Multiplying these will give: \((3x)(2x) = 6x^2\)

**Outer: **The “outer terms” are the \(3x\) and the –3. Multiplying these terms will give: \((3x)(-3) = -9x\)

**Inner:** The inner terms are the 1 and the \(2x\). Multiplying these terms will give: \((1)(2x) = 2x\)

**Last:** The last terms are the 1 and the -3. Multiplying these terms will give: \((1)(-3) = -3\)

The final answer would be the terms we got in each step above added together. The steps all together would look like this:

Apply FOIL:

\((3x+1)(2x-3) = \underbrace{(3x)(2x)}_\text{First}+\underbrace{(3x)(-3)}_\text{Outer}+\underbrace{(1)(2x)}_\text{Inner}+\underbrace{(1)(-3)}_\text{Last}\)

Simplify:

\(\begin{align} (3x)(2x)+(3x)(-3)+(1)(2x)+(1)(-3) &= 6x^2-9x+2x-3\\ &= \boxed{6x^2-7x-3}\end{align}\)

Let’s try another example, but this time one that looks a little bit different.

### Example

Multiply: \((x+y)^2\)

### Solution

If you think about what an exponent means, you can rewrite this as \((x+y)(x+y)\). Now we can apply FOIL and simplify to get the final answer:

\(\begin{align} (x+y)^2 &=(x+y)(x+y)\\ &=\underbrace{x^2}_\text{First} + \underbrace{xy}_\text{Outer} + \underbrace{yx}_\text{Inner} + \underbrace{y^2}_\text{Last}\\ &= \boxed{x^2 + 2xy + y^2}\end{align}\)

When simplifying here, remember that \(xy\) is the same as \(yx\). That’s why we combined them to get \(2xy\).

Careful!A common mistake is to write \((x + y)^2\) as \(x^2 + y^2\). This is incorrect. You cannot distribute a power. You must use FOIL.

This shows a special rule that can be used for squaring a binomial. For example, if we have \((x + 3)^2\), we could use the result in the last example and let \(y = 3\) to show:

\((x + 3)^2 = x^2 + 2(x)(3) + (3)^2\)

But, it is not necessary to memorize this since you can always just write out the binomials and apply FOIL.

## Summary

FOIL is a method for multiplying binomials, but only when you have two binomials. In cases where you have three or more, you will need to combine FOIL along with other methods. When multiplying two binomials however, you can simply remember to multiply the First terms, the Outer terms, the Inner terms, and then the Last terms. In other words, FOIL!

## Additional reading

MathBootCamps has many other articles and lessons for algebra topics. You can find those here: https://www.mathbootcamps.com/algebra-guides-and-articles/

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