FOIL Calculator

This online calculator shows steps to multiply two binomials using the FOIL method (or the FOIL mnemonic)

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Created: 2019-07-23 07:58:22, Last updated: 2021-02-18 12:13:17
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This online calculator can help you to practice the FOIL method for multiplying two binomials. It shows steps you need to perform multiplication, according to the FOIL acronym. Binomial terms are color-coded, so you can easily follow the First-Outer-Inner-Last sequence. Note that the FOIL method is a special case of a more general method for multiplying algebraic expressions using the distributive law. For general multiplication of polynomials, you can use Polynomial multiplication calculator. Method explanation can be found below the calculator.


FOIL Calculator

FOIL method

FOIL method1

The word FOIL is an acronym for the four terms of the product:

  • First ("first" terms of each binomial are multiplied together)
  • Outer ("outside" terms are multiplied—that is, the first term of the first binomial and the second term of the second)
  • Inner ("inside" terms are multiplied—the second term of the first binomial and the first term of the second)
  • Last ("last" terms of each binomial are multiplied)

The general form is:

(a+b)(c+d)=\underbrace {ac}_{{\mathrm  {First}}}+\underbrace {ad}_{{\mathrm  {Outside}}}+\underbrace {bc}_{{\mathrm  {Inside}}}+\underbrace {bd}_{{\mathrm  {Last}}}

Note that a is both a "first" term and an "outer" term; b is both a "last" and "inner" term and so forth. The order of the four terms in the sum is not important and need not match the letters' order in the word FOIL.

The word FOIL was originally intended solely as a mnemonic for high-school students learning algebra. The term appears in William Betz's 1929 text, Algebra for Today, where he states:

... first terms, outer terms, inner terms, last terms. (The rule stated above may also be remembered by the word FOIL, suggested by the first letters of the words first, outer, inner, last.)

The distributive law

The FOIL method is equivalent to a two-step process involving the distributive law:


In the first step, the (c + d) is distributed over the addition in the first binomial. In the second step, the distributive law is used to simplify each of the two terms. Note that this process involves a total of three applications of the distributive property. In contrast to the FOIL method, the distributive method can be applied easily to products with more terms such as trinomials and higher.

Reverse FOIL

The FOIL method converts a product of two binomials into a sum of four (or fewer, if they can be combined) monomials. The reverse process is called factoring or factorization. For the example of factorization, you can check our Factoring trinomials calculator.

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