Definition: FOIL is a memory aid for multiplying two binomials: multiply the First, Outer, Inner, and Last terms, then add the four products and combine like terms.
What does FOIL stand for?
First terms, Outer terms, Inner terms, Last terms. It is the distributive property applied twice, arranged so that you do not skip a product. It only applies to two binomials.
How do you use it?
For (a + b)(c + d): first is ac, outer is ad, inner is bc, last is bd. Sum them and merge the middle terms.
Example. (3x – 4)(2x + 9). First: 3x · 2x = 6x2. Outer: 3x · 9 = 27x. Inner: -4 · 2x = -8x. Last: -4 · 9 = -36. The sum is 6x2 + 27x – 8x – 36 = 6x2 + 19x – 36.
Where do students struggle?
Sign errors in the outer and inner products are the main problem, so treat each minus sign as part of its term. Another trap is trying FOIL on a trinomial times a binomial; use the distributive property or a box instead. Checking by substituting x = 1 works well: (3 – 4)(2 + 9) = -11 and 6 + 19 – 36 = -11.