Multiplying Binomials Calculator

Enter the coefficients of two binomials in the form (ax + b)(cx + d) to expand them with the FOIL method into ac·x² + (ad + bc)·x + bd.

Quick Facts

FOIL formula
(ax+b)(cx+d) = ac·x² + (ad+bc)·x + bd
Multiply First, Outer, Inner, Last terms, then combine the two x-terms.
Difference of squares
(a+b)(a−b) = a² − b²
Special case when the two binomials share terms with opposite signs.
Perfect square
(a+b)² = a² + 2ab + b²
Special case when both binomials are identical.

Your Results

Calculated
Expanded Form
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ac·x² + (ad+bc)·x + bd
x² Coefficient (A)
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A = a × c (First term)
x Coefficient (B)
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B = a×d + b×c (Outer + Inner)
Constant Term (C)
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C = b × d (Last term)

Ready

Enter the four coefficients, then press Calculate.

How Multiplying Binomials with FOIL Works

A binomial is a polynomial with exactly two terms, such as ax + b. Multiplying two binomials, (ax + b)(cx + d), means every term in the first binomial must be multiplied by every term in the second binomial, and the like terms that result are then combined. The FOIL method — First, Outer, Inner, Last — is the standard mnemonic for doing this systematically without missing a term.

The FOIL method step by step

For (ax + b)(cx + d): First multiply the first terms of each binomial, a·x and c·x, giving ac·x². Outer multiply the outermost terms, a·x and d, giving ad·x. Inner multiply the innermost terms, b and c·x, giving bc·x. Last multiply the last terms of each binomial, b and d, giving bd. Adding all four products gives ac·x² + ad·x + bc·x + bd, and since the Outer and Inner terms are both multiples of x, they combine into a single middle term: ac·x² + (ad + bc)·x + bd.

Common mistakes when multiplying binomials

  • Forgetting a term: skipping the Outer or Inner product is the most common FOIL error — each binomial has two terms, so there must be four products before combining.
  • Sign errors: a negative coefficient carries through every product it's part of; (ax − b)(cx + d) still uses FOIL, just with b treated as a negative value.
  • Combining unlike terms: only the two x-degree-1 terms (Outer and Inner) combine — the x² term and the constant term each stand alone.
  • Dropping exponents: a·x times c·x gives ac·x², not acx — multiplying two x terms adds their exponents.

Special product shortcuts

Two patterns appear often enough to memorize directly instead of running full FOIL: the difference of squares, (a + b)(a − b) = a² − b², where the Outer and Inner terms cancel out; and the perfect square, (a + b)² = a² + 2ab + b², where the Outer and Inner terms are identical and combine to 2ab. Both are just special cases of the general FOIL expansion.

Applications

Multiplying binomials is the reverse operation of factoring a quadratic, so it's used to check factoring work, to expand expressions before solving equations, and to derive quadratic models in physics and economics (such as area, revenue, or projectile-motion formulas) that start out as a product of two linear terms.

Frequently Asked Questions

What does FOIL stand for and how does it apply to multiplying binomials?
FOIL stands for First, Outer, Inner, Last — the four pairwise products you get when multiplying two binomials (ax+b)(cx+d). First: a·c·x². Outer: a·d·x. Inner: b·c·x. Last: b·d. Adding all four terms and combining the two x-terms (Outer + Inner) gives the expanded quadratic.
What is the general formula for multiplying two binomials?
For (ax + b)(cx + d), the expanded form is ac·x² + (ad + bc)·x + bd. The x² coefficient is the product of the two leading coefficients, the constant term is the product of the two constants, and the middle x coefficient is the sum of the Outer and Inner products.
How do I multiply binomials that contain subtraction or negative terms?
Treat subtraction as adding a negative: (ax − b) is the same as (ax + (−b)). Enter the negative value for that coefficient and apply FOIL normally, keeping careful track of signs — a negative times a negative gives a positive term.
What are the special product shortcuts for binomials?
Two common patterns skip full FOIL: the difference of squares (a+b)(a−b) = a² − b², and the perfect square (a+b)² = a² + 2ab + b². Both are special cases of the general FOIL expansion where terms cancel or repeat.

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