How to Use the FOIL Method
to Multiply Binomials

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Summary

The FOIL method is one of several methods that you can use to multiply polynomials. This method only works if both of the polynomials are binomials.

If you are multiplying polynomials with more than two terms, then you should use the box method or the multiplication algorithm for polynomials. If you are multiplying monomials by polynomials, then you can use the distributive property.

How to Use the FOIL Method
to Multiply Binomials

Multiply the FIRST terms of the binomials.
Multiply the OUTER terms of the binomials.
Multiply the INNER terms of the binomials.
Multiply the LAST terms of the binomials.
Combine like terms, if needed.

\(21x^2\) \(-28x\) \(-6x\) \(+8\)

\(21x^2-34x+8\)

Be sure to follow the rules for multiplying negative numbers when you multiply the coefficients of the terms. If you don't know how to multiply variables or how to combine like terms, check out the linked pages.

Example

\((3x-2)(-4x+5)\)

I'll start by multiplying the first terms of the binomials.

\((3x)(-4x)=-12x^2\)

Then I'll multiply the outer terms.

\((3x)(5)=15x\)

Next, I'll multiply the inner terms.

\((-2)(-4x)=8x\)

Finally, I'll multiply the last terms.

\((-2)(5)=-10\)

Then, I'll add all of these answers together to form a polynomial.

\(-12x^2+15x+8x-10\)

The \(15x\) and \(8x\) are like terms, so I can combine them to simplify the answer.

\(-12x^2+23x-10\)

Answer:

\((3x-2)(-4x+5)=-12x^2+23x-10\)

Why It Works

When you multiply polynomials, you have to make sure that all of the terms in the first polynomial are multiplied by all of the terms in the second polynomial.

The FOIL method is a great way to remember all the steps and make sure that the all the terms are multiplied correctly.

The FIRST and OUTER steps allow the first term in the first polynomial to be multiplied by all the terms in the second polynomial.

The INNER and LAST steps allow the last term in the first polynomial to be multiplied by all the terms in the second polynomial.

The FOIL method only works for binomials because binomials have exactly two terms. This means that the First Outer Inner Last pattern accounts for ALL of the terms in both binomials.

If you're asked to multiply trinomials or longer polynomials, the FOIL pattern does NOT work because there are extra terms that have be multiplied. In those cases, it's easier to use the box method or the multiplication algorithm.