Foil Calculator

What this calculator does

So, you want to multiply two binomials, two-term expressions like x plus 3 and x minus 5, and get the expanded result. This tool does it with the FOIL method. You enter the coefficients of the two binomials and it returns the product, fully multiplied out and tidied up.

A dropdown lets you choose plain linear binomials or ones with exponents, and the boxes adjust to match.

How to use it

1. Choose the mode: linear binomials or general binomials with exponents.
2. Enter the coefficients a, b, c, and d, and the exponents if you are in general mode.
3. Press Calculate.

What FOIL stands for

FOIL is a memory aid for multiplying two binomials, and each letter names one of the four multiplications you have to do. F is First: multiply the first term in each bracket. O is Outer: multiply the two terms on the outside. I is Inner: multiply the two terms on the inside. L is Last: multiply the last term in each bracket. Do those four multiplications, and you have covered every combination of one term from the first bracket with one term from the second. The tool carries out exactly these four products and then assembles them.

Combining the like terms

The four products are not quite the final answer, because two of them usually match. When you multiply two ordinary binomials, the Outer and Inner products are both plain multiples of the variable, so they can be added into a single middle term. That is why multiplying two binomials with an x in each gives a three-term answer, not a four-term one: the first product gives the squared term, the Outer and Inner combine into the middle term, and the Last gives the constant. The tool does this combining for you, so what you see is the result already gathered into its simplest form.

Linear and general binomials

The tool has two modes for two situations. Linear mode handles binomials of the form a times x plus b, the common case, where every term has x to the first power or no x at all. General mode allows an exponent on the x in each binomial, so you can multiply something like a times x to the n plus b by c times x to the m plus d. When the exponents differ, the Outer and Inner products land on different powers and stay as separate terms rather than combining. This wider mode lets the same FOIL idea handle binomials that are not just straight lines.

A worked example

Take 2x plus 3 times x plus 4, in linear mode. First: 2x times x is 2x squared. Outer: 2x times 4 is 8x. Inner: 3 times x is 3x. Last: 3 times 4 is 12. The Outer and Inner terms, 8x and 3x, combine into 11x. So the answer is 2x squared plus 11x plus 12.

FOIL is the distributive property twice

FOIL is not a separate rule so much as a convenient name for something you already know. It is the distributive property, the rule that multiplying by a bracket means multiplying by each term inside, applied twice over. You distribute the first bracket across the second, then collect the pieces. This is worth knowing because FOIL only works for two binomials; the moment a bracket has three or more terms, the neat four-letter pattern breaks, but the underlying distributive property carries on working. The distributive property calculator shows that more general rule in action.

Questions people ask

What does FOIL mean?

First, Outer, Inner, Last: the four multiplications needed to multiply two binomials, one term from each bracket at a time.

Why does the answer often have three terms?

Because the Outer and Inner products usually match and combine into a single middle term, turning four products into three.

What is the difference between the two modes?

Linear mode is for binomials like a times x plus b. General mode allows exponents on the x, for binomials that are not straight lines.

Does FOIL work for any expressions?

Only for two binomials. With more than two terms in a bracket, use the distributive property directly instead.

How is FOIL related to the distributive property?

It is the distributive property applied twice. FOIL is just a handy name for that specific two-binomial case.

References

On multiplying binomials. FOIL multiplies two binomials by taking the First, Outer, Inner, and Last products, then combining like terms.

1. Eric W. Weisstein, "Binomial," from MathWorld, a Wolfram resource, on the binomial, a polynomial with two terms.
2. Paul Dawkins, "Polynomials," from Paul's Online Notes, Lamar University, on the FOIL method for multiplying two binomials.