FOIL Method Calculator

The FOIL Method

FOIL is an acronym for multiplying two binomials, meaning expressions with exactly two terms. It stands for First, Outer, Inner, Last: the four products you compute and then combine.

For (ax + b)(cx + d):

Combined result:

(ax + b)(cx + d) = (a·c)x² + (a·d + b·c)x + (b·d)

Example:

(2x + 3)(x − 5)

• First: 2 × 1 = 2 → 2x²
• Outer: 2 × (−5) = −10 → −10x
• Inner: 3 × 1 = 3 → 3x
• Last: 3 × (−5) = −15

Combined: 2x² + (−10 + 3)x + (−15) = 2x² − 7x − 15

Special cases: patterns to recognize:

Why FOIL matters:

FOIL is the foundation of polynomial multiplication. Understanding it makes factoring (the reverse operation) much easier. Factoring is essential for solving quadratic equations and simplifying rational expressions.

Entering negative coefficients:

Use negative values for b or d to represent subtraction. For (3x − 4), enter a = 3 and b = −4.

Reading the answer backwards. The most useful thing about the expanded form is that you already know its roots without touching the quadratic formula. (2x + 3)(x − 5) is zero exactly when one of the brackets is zero, so x = −3/2 or x = 5. That is why factoring is worth the effort: a factored quadratic hands you the solutions, while the expanded one makes you work for them. The calculator prints both roots and the vertex alongside the expansion so you can see the connection.

A limit worth knowing. FOIL only applies to two binomials. Multiplying a binomial by a trinomial, or two trinomials, needs the general distributive rule: every term in the first bracket times every term in the second. FOIL is that same rule with the four products named for a case people meet constantly. Students who learn FOIL as a rule rather than as a shortcut tend to get stuck the first time a bracket has three terms in it.