Multiplying Binomials Calculator — FOIL Method
Expand (ax + b)(cx + d) using the FOIL method — First, Outer, Inner, Last — and combine like terms to get the trinomial result in seconds.
First — leading terms × leading terms
Outer — first leading × second constant
Inner — first constant × second leading
Last — constant × constant
Combine like x-terms (Outer + Inner)
Expanded result
- 1
First: (ax)(cx) = acx²
1 × 1 = 1 - 2
Outer: (ax)(d)
1 × -2 = -2 - 3
Inner: (b)(cx)
3 × 1 = 3 - 4
Last: (b)(d)
3 × -2 = -6 - 5
Linear coefficient: outer + inner
-2 + 3 = 1The two x-terms are like terms and combine into a single coefficient.
How does this calculator work?
Multiply (ax + b)(cx + d) with FOIL: First = acx², Outer = adx, Inner = bcx, Last = bd. Combine the outer and inner x-terms to get (ad + bc)x. Result: acx² + (ad + bc)x + bd. The middle coefficient is the sum of the two cross products; the constant is the product of the two constants.
Formula
How this is calculated
The FOIL method is a mnemonic for applying the distributive property when multiplying two binomials. Each letter stands for one of four term-pairs to multiply: First (the two leading terms ax and cx), Outer (the outermost pair ax and d), Inner (the innermost pair b and cx), and Last (the two constants b and d). Their four products are F = acx², O = adx, I = bcx, and L = bd.
After computing the four partial products, the two x-terms O and I are like terms and combine as (ad + bc)x. The final expanded form is therefore acx² + (ad + bc)x + bd — a trinomial with at most three distinct terms. If ac = 0 the quadratic term vanishes; if ad + bc = 0 the linear term cancels; if bd = 0 there is no constant. Any of the four coefficients can be negative or zero.
For the special case a = c = 1, the result is x² + (b + d)x + bd, so the middle coefficient equals the sum of the two constants and the constant term equals their product. This identity is the basis for factoring monic quadratics: given x² + px + q, find two numbers that add to p and multiply to q, and the binomials are (x + r)(x + s) where r + s = p and r · s = q.
Frequently asked questions
FOIL is a mnemonic for the four products that arise when you multiply two binomials: First (leading terms), Outer (outermost terms), Inner (innermost terms), Last (constant terms). Adding all four and combining like terms gives the complete expansion.
Yes. For the perfect square (x + b)², set a = c = 1 and d = b — the result is x² + 2bx + b². For the difference of squares (x + b)(x − b), set a = c = 1 and d = −b — the linear term cancels and you get x² − b².
For a monic trinomial x² + px + q, find two numbers r and s where r + s = p and r · s = q. Then x² + px + q = (x + r)(x + s). Set a = c = 1, b = r, d = s in this calculator to verify the expansion matches your original trinomial.
Also known as
TG we-Calculate Editorial Team. (2026). Multiplying Binomials Calculator — FOIL Method [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/multiplying-binomials-calculator
TG we-Calculate Editorial Team. "Multiplying Binomials Calculator — FOIL Method." TG we-Calculate. 2026. https://we-calculate.com/calculator/multiplying-binomials-calculator.
TG we-Calculate Editorial Team, "Multiplying Binomials Calculator — FOIL Method," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/multiplying-binomials-calculator
@misc{wecalculate_multiplying_binomials_calculator, title = {Multiplying Binomials Calculator — FOIL Method}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/multiplying-binomials-calculator}}, year = {2026}, note = {TG we-Calculate} }
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