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.
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
Multiply:
\[(6x-7)(-2x+5)\]
What is the product of the FIRST terms?
\[({\red 6x}-7)({\red -2x}+5)\]
\[({\red 6x})({\red -2x})={\red -12x^2}\]
What is the product of the OUTER terms?
\[({\yellow 6x}-7)(-2x{\yellow +5})\]
\[({\yellow 6x})({\yellow +5})={\yellow +30x}\]
What is the product of the INNER terms?
\[(6x{\green -7})({\green -2x}+5)\]
\[({\green -7})({\green -2x})={\green +14x}\]
What is the product of the LAST terms?
\[(6x{\blue -7})(-2x{\blue +5})\]
\[({\blue -7})({\blue +5})={\blue -35}\]
Are there any like terms?
\[(6x-7)(-2x+5)={\red -12x^2}{\yellow +30x}{\green +14x}{\blue -35}\]
Yes, \({\yellow +30x}\) and \({\green +14x}\) are like terms so I can add them together.
\[{\yellow +30x}{\green +14x}=+44x\]
When you multiply \((6x-7)(-2x+5)\), the expanded answer is…
\[{\red -12x^2}+44x{\blue -35}\]
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.
