What does FOIL stand for?

FOIL stands for First, Outer, Inner, Last—the order in which you multiply the terms of two binomials.

Does the FOIL method work with fractions and decimals?

Yes. FOIL is an application of the distributive property and works with integers, decimals, and fractions.

What if one binomial has a negative sign?

Include the sign with the coefficient. The method is the same; signs are carried through during multiplication.

Is FOIL the same as distribution?

Yes. FOIL is a mnemonic for the distributive property applied to two-term polynomials (binomials).

Can I use variables other than x?

Yes. This tool lets you choose any single letter for the variable, such as y or z.

Why do the inner and outer terms sometimes cancel?

If ad + bc = 0, the x-term cancels out, leaving only the quadratic and constant terms.

FOIL Method Calculator

Use this professional FOIL calculator to expand the product of two binomials and see every step: First, Outer, Inner, Last. It’s designed for students, educators, and professionals who need fast, accurate algebra with support for integers, decimals, and fractions.

Calculator

Variable

First binomial: coefficient of variable (a) *

First binomial: constant term (b) *

Second binomial: coefficient of variable (c) *

Second binomial: constant term (d) *

Preview

(1x + 2)(3x + 4)

Results

First (a·c) term

—

Outer (a·d) term

—

Inner (b·c) term

—

Last (b·d) term

—

Expanded Polynomial

Enter values and click Calculate to see the LaTeX expansion.

Data Source and Methodology

Authoritative reference: OpenStax, Algebra and Trigonometry 2e (2022), Section 5.2 “Multiplying Polynomials”. Available at: openstax.org/details/books/algebra-and-trigonometry-2e.

All calculations are derived from the distributive property applied to binomials (FOIL).

Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.

The Formula Explained

Given two binomials in a single variable, \((a\,x + b)(c\,x + d)\), the FOIL method applies the distributive property to obtain:

First: \((a\,x)\cdot(c\,x) = a c\,x^{2}\)
Outer: \((a\,x)\cdot d = a d\,x\)
Inner: \(b\cdot(c\,x) = b c\,x\)
Last: \(b\cdot d = b d\)

Combining like terms yields the standard form:
\((a\,x + b)(c\,x + d) = a c\,x^{2} + (a d + b c)\,x + b d.\)

Glossary of Variables

a — coefficient of the variable in the first binomial.
b — constant term in the first binomial.
c — coefficient of the variable in the second binomial.
d — constant term in the second binomial.
x — the chosen variable; you may use any single letter such as x, y, or z.
First, Outer, Inner, Last — the four products computed during expansion.
Expanded polynomial — the simplified result in standard form: quadratic term, linear term, constant.

Come Funziona: Un Esempio Passo-Passo

Example: Expand \((2x + 3)(x - 5)\).

First: \((2x)(x) = 2x^{2}\)
Outer: \((2x)(-5) = -10x\)
Inner: \(3(x) = 3x\)
Last: \(3(-5) = -15\)
Combine: \(2x^{2} - 10x + 3x - 15 = 2x^{2} - 7x - 15\).

Frequently Asked Questions (FAQ)

What is the FOIL method?

FOIL is a mnemonic for multiplying two binomials: First, Outer, Inner, Last. It is simply the distributive property applied in a helpful order.

Can I enter fractions or decimals?

Yes. Enter integers, decimals (e.g., 1.25), or fractions (e.g., -3/5). The calculator keeps exact fractional arithmetic internally.

What if a coefficient is 1 or -1?

The tool handles 1 and -1 correctly. For example, 1x is shown as x and -1x as -x in the preview and results.

Will the linear term sometimes disappear?

Yes. If the sum of the Outer and Inner coefficients equals zero (ad + bc = 0), the x term cancels out and the result has no linear term.

Is FOIL applicable to polynomials with more than two terms?

FOIL is specific to binomials. For larger polynomials, use general distribution or area models (box method).

Which variable can I choose?

Any single letter. This tool defaults to x but supports y, z, or any alphabetic character.

Tool developed by Ugo Candido. Content verified by CalcDomain Math Editorial Team.
Last reviewed for accuracy on: September 14, 2025.