FOIL Calculator

FOIL is an acronym for First, Outer, Inner, Last. It describes the order in which you multiply terms when expanding the product of two binomials: (a + b)(c + d). First means multiply a times c, Outer means a times d, Inner means b times c, and Last means b times d.

Step 1: Multiply the First terms (first term of each binomial). Step 2: Multiply the Outer terms (first of first, last of second). Step 3: Multiply the Inner terms (last of first, first of second). Step 4: Multiply the Last terms (last term of each). Step 5: Add all four products. Step 6: Combine like terms to simplify.

FOIL specifically works for multiplying two binomials (expressions with exactly two terms each). For multiplying a binomial by a trinomial, or two trinomials, you need to use the full distributive property, multiplying each term in the first polynomial by each term in the second. FOIL is really just a mnemonic for the distributive property applied to binomials.

Three important patterns: (1) Perfect square: (a + b)^2 = a^2 + 2ab + b^2, (2) Difference of squares: (a + b)(a - b) = a^2 - b^2, (3) Standard trinomial: (x + a)(x + b) = x^2 + (a+b)x + ab. Recognizing these patterns speeds up multiplication and factoring.

The Outer and Inner products often contain like terms (same variable to the same power). For example, in (x + 3)(x + 2), the Outer is 2x and Inner is 3x. Since both are "x to the first power" terms, they combine to give 5x. This is why the result is usually a trinomial rather than four terms.

FOIL is really just a specific application of the distributive property. When you distribute (a + b)(c + d), you get a(c + d) + b(c + d) = ac + ad + bc + bd. This is exactly what FOIL produces. FOIL is simply a mnemonic to help remember all four multiplications.

The reverse of FOIL is factoring. If you have a trinomial like x^2 + 5x + 6, you can factor it back into (x + 2)(x + 3). To factor x^2 + bx + c, find two numbers that multiply to c and add to b. Those become the constants in your binomial factors.

Yes, FOIL works with any numbers including negatives. Just be careful with signs. For example, (x - 3)(x + 2): First = x^2, Outer = 2x, Inner = -3x, Last = -6. Result: x^2 + 2x - 3x - 6 = x^2 - x - 6. Remember that minus times plus gives minus.