The **FOIL Method** is used to multiply binomials. $FOIL$ is an acronym. The letters stand for First, Outside, Inside, and Last, referring to the order of multiplying terms. You multiply first terms, then outside terms, then inside terms, then last terms, and then combine like terms for your answer.

The mnemonic $FOIL$ tells us exactly what terms to multiply, and in what order:

**F**irst – multiply the*first*terms**O**utside – multiply the*outside/outer*terms**I**nside – multiply the*inside/inner*terms**L**ast – multiply the*last*terms

*[Insert a large, colorful graphic showing the problem (5n + 3)(n + 6), with identifying arrows and words pointing to the 5n and n as First; the 5n and 6 as Outside; the 3 and n as Inside; and the 3 and 6 as Last]*

By following First, Outer, Inner, Last, we do not overlook any term in either binomial. All the terms in the first binomial are combined with the terms in the second binomial properly. Everything gets counted.

Let's apply the $FOIL$ method on a couple of examples.

Here we are multiplying two binomials:

$\left(q-3\right)\left(q-7\right)$

Let's go through each step of $FOIL$ to solve this multiplication problem:

**F**irst, multiply first terms of each binomial: $q*q={\mathit{q}}^{\mathbf{2}}$**O**utside terms are multiplied next: $q*(-7)=\mathbf{-}\mathbf{7}\mathit{q}$**I**nside terms are multiplied next: $-3*q=\mathbf{-}\mathbf{3}\mathit{q}$**L**ast, multiply last terms of each binomial: $-3*(-7)=\mathbf{21}$

Put your four answers down on paper in the order you found them: ${q}^{2}-7q-3q+21$

Finish by combining like terms: ${\mathit{q}}^{\mathbf{2}}\mathbf{}\mathbf{-}\mathbf{}\mathbf{10}\mathit{q}\mathbf{}\mathbf{+}\mathbf{}\mathbf{21}$

Now, let's use the $FOIL$ method on this equation:

$\left(5n+3\right)\left(n+6\right)$.

Go through each letter:

**F**irst – $5n*n=5{n}^{2}$**O**utside – $5n*6=30n$**I**nside – $3*n=3n$**L**ast – $3*6=18$

Put your answers together: $5{n}^{2}+30n+3n+18$

Combine like terms: $\mathbf{5}{\mathit{n}}^{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{33}\mathit{n}\mathbf{}\mathbf{+}\mathbf{}\mathbf{18}$

Our final answer, the product of two binomials, contains three terms so it is a trinomial.

Multiplying three binomials is a special case for $FOIL$ because the $FOIL$ method can only be used for multiplying two binomials at a time.

You can use FOIL to multiply three or more binomials if you pair them off, then factor the answer to the remaining binomial.

FOIL cannot be used for binomial addition, subtraction, or division.

Here is a multiplication equation with three binomials:

$\left(4y-7\right)\left(2y+10\right)\left(7-2\right)$

To begin, we pair off the first two binomials:

$\left(4y-7\right)\left(2y+10\right)$

Then, multiply them using the $FOIL$ method, and we get:

$\left(8{y}^{2}+40y-14y-70\right)$

Next, we combine like terms:

$\left(8{y}^{2}+26y-70\right)$

Now, we multiply our new binomial with the remaining binomial form the original equation:

$\left(8{y}^{2}+26y-70\right)\left(y-2\right)$

Then, factoring and simplifying is the final step:

$\left(\mathbf{8}{\mathit{y}}^{\mathbf{3}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{10}{\mathit{y}}^{\mathbf{2}}\mathbf{}\mathbf{-}\mathbf{}\mathbf{122}\mathit{y}\mathbf{}\mathbf{+}\mathbf{}\mathbf{140}\right)$

If you are faced with more multiplying two binomials, solve two at a time using $FOIL$ until you are left with just one polynomial.

The term **polynomial** refers to an expression of constants, variables, and exponents that are added, subtracted, or multiplied, like the highlighted answer above. Each term on it's on is called a monomial.

Give these practice problems a try.

- What does $FOIL$ stand for in math?
- What is the $FOIL$ method in mathematics?
- Apply the $FOIL$ method in math to this problem: $\left(2x+13\right)\left(2x-17\right)$

Don't *foil* around with partial work; get answers down on paper before you check our answers.

- $FOIL$ stands for First, Outside, Inside, and Last. It is a mnemonic way to multiply two binomials.
- The $FOIL$ method in mathematics allows you to multiply two binomials quickly and helps to ensure you miss no part of the problem and gather all the partial products.
- To apply the $FOIL$ method in math to this problem: $\left(2x+13\right)\left(2x-17\right)$, we would write:

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

$2x*2x=4{x}^{2}$

**O**utside terms are multiplied next:

$2x*(-17)=-34x$

**I**nside terms are multiplied next:

$13*2x=26x$

**L**ast, multiply last terms of the binomials:

$13*(-17)=-221$

Now we have:

$4{x}^{2}-34x+26x-221$

Combining like terms yields:

$\mathbf{4}{\mathit{x}}^{\mathbf{2}}\mathbf{}\mathbf{-}\mathbf{}\mathbf{8}\mathit{x}\mathbf{}\mathbf{-}\mathbf{}\mathbf{221}$

After working your way through this lesson and video, you will have learned:

- What FOIL means in math
- How to use foil to solve multiplication of binomials
- Foiling three binomials

Instructor: **Malcolm M.**

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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