How this instrument works
FOIL is a mnemonic, not a separate law of algebra — it names the four multiplications produced when the distributive property is applied twice to (ax + b)(cx + d). First distribute the whole second binomial across ax, then across b: ax(cx + d) + b(cx + d) = acx² + adx + bcx + bd. Group those four terms by name — First (ac), Outer (ad), Inner (bc), Last (bd) — and FOIL is exactly that expansion with labels attached so none of the four gets skipped.
The acronym only fits the case it was built for: two terms times two terms. Multiply a trinomial by a binomial and you get six products, not four, and FOIL has no letter for the extra two — the distributive property still works, but the four-letter shortcut stops applying the moment either factor grows past two terms. That narrowness is the trade for its usefulness: within its own case, FOIL fixes an order to walk through so a cross-term is never quietly dropped.
A genuinely useful edge case shows up when the two binomials share a leading coefficient but carry opposite constants, as in (x − k)(x + k). Outer becomes kx and Inner becomes −kx, and the two cancel exactly, leaving only x² − k² — the difference-of-squares identity emerging as a special case where FOIL's usual four terms collapse to two. Square a binomial instead, (x + k)(x + k), and Outer and Inner come out equal rather than opposite, doubling into 2kx instead of vanishing.
- Enter the first binomial's terms into a, in (ax + b) and b, in (ax + b) — for (2x + 3) that is a = 2 and b = 3.
- Enter the second binomial's terms into c, in (cx + d) and d, in (cx + d).
- Read x² coefficient (First) directly — it is the product ac and needs no further work.
- Add x coefficient from Outer to x coefficient from Inner yourself; the sheet reports ad and bc separately so each cross-term can be checked before they are combined into the single middle-term coefficient.
- Read Constant (Last) for the final term of the expansion, the product bd.
Worked example — expanding (2x + 3)(4x + 5)
Enter a = 2 and b = 3 for the first binomial, (2x + 3), then c = 4 and d = 5 for the second, (4x + 5). The sheet returns x² coefficient (First) = 8, from 2x times 4x; x coefficient from Outer = 10, from 2x times 5; x coefficient from Inner = 12, from 3 times 4x; and Constant (Last) = 15, from 3 times 5 — four separate multiplications, none harder than a single-digit fact.
Add the two middle terms by hand: Outer 10 plus Inner 12 makes 22, so (2x + 3)(4x + 5) expands to 8x² + 22x + 15. A quick check confirms it without redoing the algebra — plug x = 1 into both sides: the original factors give 5 × 9 = 45, and the expansion gives 8 + 22 + 15 = 45, the same figure either way.
Questions
What does FOIL stand for?
First, Outer, Inner, Last — the four pairings needed to multiply two binomials, (ax + b)(cx + d). First multiplies the two leading terms, ax and cx; Outer multiplies the two outermost terms, ax and d; Inner multiplies the two innermost terms, b and cx; and Last multiplies the two remaining constants, b and d. Adding all four gives the full expansion.
Why does FOIL actually work?
It is the distributive property applied twice, not a rule of its own. Expanding (ax + b)(cx + d) as (ax)(cx + d) + b(cx + d), then distributing each of those two pieces again, produces exactly the four terms acx², adx, bcx, and bd — FOIL simply names the order so none of the four is skipped when doing it by hand.
Why do the Outer and Inner terms sometimes cancel out?
They cancel when the two binomials share a leading coefficient but carry opposite constants, such as (x − k)(x + k). Outer becomes kx and Inner becomes −kx, so the two sum to zero and only x² − k² survives — the difference-of-squares identity, one of the most common patterns a FOIL expansion can collapse into.
What is the most common mistake when applying FOIL?
Dropping the Inner term. First and Last sit at the visual edges of the two parentheses and are easy to spot; Outer and Inner both touch the middle of the expression, and Inner — the product of the two innermost terms — is the one most often left out, since it runs against the eye's natural left-to-right sweep.
Does FOIL work for more than two binomials, or for a trinomial?
No — FOIL only names the four products that appear when two two-term expressions are multiplied. A trinomial times a binomial produces six terms, not four, and needs the distributive property applied across every pairing rather than the FOIL acronym, which was built specifically around the two-by-two case.
Why does this sheet report Outer and Inner separately instead of already combined?
So each cross-multiplication can be checked on its own before they are added. The coefficient of x in the final answer is always Outer plus Inner, since both are ordinary like terms in x — but a wrong combined total is hard to trace, while a wrong Outer or Inner value, shown separately, points straight at which single multiplication went wrong.