1Cademy - The FOIL Method for Multiplying Binomials

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Multiplying a Binomial by a Binomial Using the Distributive Property

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The FOIL Method for Multiplying Binomials

The FOIL method is a mnemonic shortcut for multiplying two binomials that organizes the same four products produced by two rounds of the Distributive Property into a memorable sequence. The letters in FOIL stand for:

First — Multiply the first term of each binomial.
Outer — Multiply the two outermost terms (the first term of the first binomial and the last term of the second binomial).
Inner — Multiply the two innermost terms (the last term of the first binomial and the first term of the second binomial).
Last — Multiply the last term of each binomial.

For example, applying FOIL to $(x - 2)(x - y)$ :

F: $x \cdot x = x^2$
O: $x \cdot (-y) = -xy$
I: $(-2) \cdot x = -2x$
L: $(-2) \cdot (-y) = 2y$

This gives $x^2 - xy - 2x + 2y$ . The FOIL pattern guarantees that all four products are found. After obtaining the four terms, the final step is to combine any like terms. When both binomials involve a single variable, the Outer and Inner products are typically like terms that combine into one term, yielding a trinomial. When the binomials involve different variables, there may be no like terms to combine, leaving four terms in the result.

FOIL produces exactly the same result as the Distributive Property approach — the two methods are equivalent. FOIL simply provides an easy-to-remember word that ensures no product is missed.

Important limitation: The FOIL method applies only when multiplying two binomials — it cannot be used with other types of polynomials. For products that involve non-binomial factors (such as a binomial times a trinomial), the general Distributive Property approach must be used instead. A practical tip when applying FOIL is to draw lines connecting each pair of terms being multiplied; these visual arrows reinforce the F-O-I-L sequence and make it easier to ensure that no product is overlooked.

Updated 2026-05-01

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