FOIL Calculator (multiply two binomials: first, outer, inner, last)

Understanding the FOIL calculator — concept and principle

This calculator multiplies two binomials (ax + b)(cx + d) with the FOIL method: First, Outer, Inner, Last. It shows each of the four products, the combined middle term, the final trinomial and a check at x = 1. Use it to learn or check binomial multiplication.

First, Outer, Inner, Last: every term of one bracket meets every term of the other.

How it is calculated

F · First: ax · cx = acx²

O · Outer: ax · d = adx

I · Inner: b · cx = bcx

L · Last: b · d = bd

Result: acx² + (ad + bc)x + bd

Check: put x = 1 into both sides; (a + b)(c + d) must equal ac + ad + bc + bd.

The principle

FOIL is the distributive property used twice: each term of the first binomial multiplies each term of the second, giving four products. The outer and inner products both contain x, so they are like terms and combine into the middle term. The word FOIL is just a memory aid for making sure none of the four products is missed; the box method shows the same four products as areas.

Worked example

Default inputs: (2x + 3)(x − 5), i.e. a = 2, b = 3, c = 1, d = −5.

First: 2x · x = 2x².
Outer: 2x · (−5) = −10x; Inner: 3 · x = 3x.
Last: 3 · (−5) = −15.
Middle term: −10x + 3x = −7x.
Product: 2x² − 7x − 15; check at x = 1: (2 + 3)(1 − 5) = −20, and 2 − 7 − 15 = −20.

How to read the result

The leading coefficient is the product of the two x-coefficients, the constant is the product of the two constants, and the middle coefficient is ad + bc. Reading this in reverse is how trinomials are factored. If the check at x = 1 fails, look first at the signs of the outer and inner products.

Common mistakes

Forgetting the outer or inner product.
Dropping the negative sign of d.
Combining x² and x terms.
Using FOIL for longer polynomials, where every term must still multiply every term.

Keep in mind: FOIL only covers two binomials. For a binomial times a trinomial, use the full distributive property or the box method.

Frequently asked questions

What does FOIL stand for?

First, Outer, Inner, Last: the four products when multiplying two binomials.

Is FOIL the same as the distributive property?

Yes, it is the distributive property applied twice.

How do I check a FOIL answer?

Substitute a value such as x = 1 into both the product and the expanded result.

Sources: OpenStax, Elementary Algebra 2e (2020), §6.3 Multiply Polynomials

Checked 9 Nov 2026. OpenStax, Elementary Algebra 2e (2020), §6.3 Multiply Polynomials — “Multiply a Binomial by a Binomial Using the FOIL Method” (First, Outer, Inner, Last).

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