Multiplying Binomials Calculator - FOIL Method

A binomial is a polynomial that contains exactly two terms joined by either addition or subtraction. Examples include (x + 3), (2y − 7), and (5a + 1). Multiplying two binomials produces a four-term intermediate expression that simplifies into a trinomial after combining like terms. This operation is one of the most foundational skills in algebra and underlies factoring, quadratic equations, and polynomial arithmetic throughout mathematics. The FOIL method is the standard mnemonic for multiplying two binomials. FOIL stands for First, Outer, Inner, Last — the four pairs of terms that must be multiplied together when expanding (ax + b)(cx + d). The First step multiplies the leading terms: ax × cx = acx². The Outer step multiplies the first term of the first binomial by the last term of the second: ax × d = adx. The Inner step multiplies the second term of the first binomial by the first term of the second: b × cx = bcx. The Last step multiplies the two trailing constants: b × d = bd. After collecting all four products, the Outer and Inner terms both contain x, so they combine into (ad + bc)x, yielding the standard trinomial acx² + (ad + bc)x + bd. FOIL is really just the distributive property applied twice. Writing ax(cx + d) + b(cx + d) makes the logic explicit: each term of the first binomial distributes across the entire second binomial. This perspective is important because it explains how to multiply longer polynomials — a trinomial multiplied by a binomial requires distributing all three terms of the trinomial over the binomial, producing six intermediate products rather than four. Several special products follow predictable patterns that are worth recognising. The difference of squares, (a + b)(a − b), always collapses to a² − b² because the outer and inner terms cancel. A perfect square trinomial, (a + b)², expands to a² + 2ab + b², where the middle term is twice the product of the two constants. Knowing these shortcuts speeds up mental arithmetic and makes factoring significantly easier, because factoring is simply the inverse of expansion. Practical applications of binomial multiplication arise across many fields. In geometry, if the length and width of a rectangle are both expressed as binomials, the area is found by multiplying them. In physics and engineering, kinematic equations for displacement and quadratic models for projectile paths often require expanding binomial expressions. In finance, compound-interest approximations for small rates use binomial expansion. Mastering this calculation builds the algebraic fluency needed for completing the square, working with the quadratic formula, and eventually tackling polynomial calculus.