FOIL Method
FOIL is a mnemonic for multiplying two binomials: First, Outside, Inside, Last, standing for the four pairs of terms to multiply.
Formula
(a+b)(c+d) = ac + ad + bc + bd
Definition
FOIL is a mnemonic for multiplying two binomials $(a + b)(c + d)$: multiply the First terms ($ac$), Outer terms ($ad$), Inner terms ($bc$), and Last terms ($bd$), then combine like terms. It is simply a notational shortcut for the distributive law in the ring $F[x]$, and it does not generalize directly to products of three or more binomials or to polynomials with more than two terms, those require the full Cauchy product (distributive) approach.
Example
$(x + 3)(x + 7)$: First $x \cdot x = x^2$, Outside $x \cdot 7 = 7x$, Inside $3 \cdot x = 3x$, Last $3 \cdot 7 = 21$; adding gives $x^2 + 10x + 21$. $(2x - 5)(3x + 4)$: F $6x^2$, O $8x$, I $-15x$, L $-20$, summing to $6x^2 - 7x - 20$. A product like $(a+b)(c+d)(e+f)$ requires applying FOIL to two of the factors first, then distributing the result over the third, giving up to $8$ terms before combining.
Key Insight
FOIL only works for exactly two binomials; for bigger polynomials, the full distributive property is needed. FOIL's four terms often reduce to three once the outer and inner products combine (they are like terms), and recognizing when they cancel (difference of squares) or double (perfect square trinomial) saves steps. FOIL is really just a pedagogical device encoding the distributive law for one specific case, understanding why it works, because multiplication distributes over addition in any ring, gives the general principle behind every polynomial product.