FOIL Method Calculator

How do I enter a binomial such as 3x − 2?

Put the sign inside the number: a = 3 and b = −2 for 3x − 2. The form multiplies (ax + b)(cx + d), so subtraction is just a negative b or d. For (x + 3)(x − 3) enter a = 1, b = 3, c = 1, d = −3.

What does FOIL stand for, and why does it work?

First, Outer, Inner, Last: the four products ac·x², ad·x, bc·x and bd. It is the distributive law applied twice, which is why the result is acx² + (ad + bc)x + bd. The Outer and Inner terms both contain x, so they are added into one middle coefficient.

How do I enter (x + 2)(x + 5), where x has no visible coefficient?

Enter 1 for each x coefficient (a = 1, c = 1). A coefficient left at 0 removes the x term entirely, so entering a = 0 turns the first binomial into the constant b and the product becomes linear. For example a = 0, b = 2, c = 3, d = 5 gives 6x + 10.

Why can the x² term disappear from the answer?

The x² coefficient is a × c, so it is zero whenever a or c is zero. The expanded line then drops that term. If every product is zero, it prints 0.

Does it work with decimals and fractions?

Decimals work, for example a = 0.5, b = 2, c = 4, d = −3 gives 2x² + 6.5x − 6. Fractions have to be converted to decimals first, and results are rounded to six decimal places. A coefficient such as ⅓ becomes 0.333333, so the output is approximate, not exact.

What shortcuts does FOIL contain?

Three patterns appear often: (x + p)² = x² + 2px + p², (x − p)² = x² − 2px + p², and (x + p)(x − p) = x² − p². In the second case the middle term cancels: a = 1, b = 3, c = 1, d = −3 gives x² − 9 with an x coefficient of 0.

Does it multiply bigger polynomials, like a trinomial times a binomial?

No. FOIL only describes two binomials. A trinomial times a binomial needs each term of the first multiplied by each term of the second, six products in all. This form has fields for two binomials in one variable and nothing else.

Can I check my homework answer with it?

Yes. The page lists each of the four products separately, labelled FIRST, OUTER, INNER and LAST, so you can compare your own steps rather than just the final line. Note that the working lines show a coefficient of 1 as "1x²", while the final polynomial writes it as "x²".