FOIL Calculator
This FOIL calculator multiplies two binomials and names all four products — First, Outer, Inner and Last — before it collects anything.
Type (x + 3)(x - 5) the way it is printed in your book. No coefficient boxes.
It is the quickest way to multiply two binomials by hand, and the first outer inner last order is what stops you missing one of the four.
What FOIL Stands For
First, Outer, Inner, Last. The FOIL method is four multiplications in that order, and every one of them is a pair of terms:
First — the first term of each bracket
Outer — the two on the outside, far left and far right
Inner — the two in the middle, sitting next to each other
Last — the last term of each bracket
The word was coined by William Betz in his 1929 textbook Algebra for Today, purely as something for students to remember. It has outlasted the book by nearly a century.
Worked Through, Line by Line
Take (2x + 1)(3x − 4):
| Step | Which two terms | Product |
|---|---|---|
| First | 2x · 3x | 6x² |
| Outer | 2x · (−4) | −8x |
| Inner | 1 · 3x | 3x |
| Last | 1 · (−4) | −4 |
Written out: 6x² − 8x + 3x − 4. Then O and I collect, because −8x and 3x are the same kind of term, and the answer is 6x² − 5x − 4.
Notice that F and L never collect with anything. Only the middle pair ever meet, which is why the answer usually has three terms rather than four.
Why Four Boxes Are the Wrong Way to Ask
Look at what the other tools want from you. One asks for "the four coefficients in their respective fields". Another wants the question rewritten as (ax + b)(cx + d) before it will look at it. A third asks for the coefficients and the variables separately.
So before any of them helps, you have to pull the question apart and decide which number is a and which is b — with the minus signs. That is precisely the step where answers go wrong, and it is the step they hand back to you.
This page reads both brackets as written. One box, and you type what is on the page in front of you.
That is the whole of the difference, and it is not a small one. Every sign error in this topic starts somewhere, and for most students it starts before the multiplying does — in the moment they decide, from a printed question, which number goes in which box and whether the minus travels with it.
A Squared Bracket Is Two Brackets
(x − 7)² catches more students than anything else on this page.
It is not x² + 49. Squaring a bracket does not mean squaring the pieces — it means writing the bracket down twice and multiplying: (x − 7)(x − 7). Now FOIL applies, O and I are both −7x, and the answer is x² − 14x + 49.
That −14x is the term that disappears when people take the shortcut. Type a square into the calculator above and it writes the two brackets out first, on purpose.
When the Middle Term Vanishes
(x + 3)(x − 3) gives x² − 9. No middle term at all.
It is not a special rule to memorise. The Outer product is −3x, the Inner is +3x, and they cancel each other out completely, leaving nothing in the middle to write down.
This happens whenever the two brackets are identical apart from one sign, and the answer is always one square take another. It is worth learning to spot on sight, because once you can, a whole family of questions stops needing any working at all — and the same pattern turns up again the moment you start factorising, running backwards.
Signs Are Where the Marks Go
The minus belongs to the term, not to the operation.
In (x − 7), the second term is negative seven — so Last is (−7) · (−7) = +49, positive. Not −49, which is what comes out if you treat the minus as an instruction to subtract at the end rather than as part of the number you are multiplying.
Two negatives making a positive is the rule everybody knows and half of everybody forgets under exam pressure. The calculator shows each product with its own sign so you can check yours against it one at a time rather than hunting through a finished answer.
What FOIL Cannot Do
FOIL is a mnemonic for two brackets with two terms each. That is all it was ever meant to be.
You can expand binomials of any shape this way, but only two-by-two earns the name. Put three terms in a bracket and the four letters run out — there are six products, not four. The method behind it still works, because every term of the first bracket must meet every term of the second, but the name no longer fits. For those, the multiplying polynomials calculator lays out a grid instead.
Going the other way — starting from x² − 2x − 15 and finding the brackets — is a different skill again, and the factoring trinomials calculator handles that one.
And once the brackets are gone, the usual next line of the question is a number: what does x² − 2x − 15 come to when x = 4? That is a substitution rather than an expansion, and the evaluating expressions calculator takes it from there.
How to Use This FOIL Calculator
Type both brackets: (x + 3)(x - 5). A square works as (x - 7)^2 and is expanded for you. Coefficients need no times sign, so (2x + 1)(3x - 4) reads on its own, and two letters are fine — (x + y)(x - y).
The pad carries x and y; any other letter comes from your own keyboard. Fractions such as (x/2 + 1) stay exact rather than turning into decimals.
FOIL Calculator FAQ
What does FOIL stand for?
Which terms are the outer and inner ones?
Does FOIL work on three terms?
How do I square a bracket?
Why did my middle term disappear?
What happens to the signs?
Is FOIL the same as the distributive property?
Do I always have to use FOIL?
Four multiplications, written down before any of them are added. F and L stand alone, O and I usually meet, and the signs travel with the terms rather than with the operations.
