Multiplying Binomials Calculator

Enter two binomial expressions like (ax + b) and (cx + d) using the input fields below. The Multiplying Binomials Calculator applies the FOIL method (First, Outer, Inner, Last) to expand the product, showing the expanded polynomial result and each intermediate FOIL step so you can follow the algebra.

The coefficient of x in (ax + b)

The constant term b in (ax + b)

The coefficient of x in (cx + d)

The constant term d in (cx + d)

Results

Expanded Result

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F — First Terms (x² coefficient)

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O — Outer Terms (x coefficient part 1)

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I — Inner Terms (x coefficient part 2)

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L — Last Terms (constant)

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Combined x Coefficient (Outer + Inner)

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Polynomial Expression

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FOIL Term Contributions

Results Table

Frequently Asked Questions

What is the FOIL method?

FOIL is a mnemonic for multiplying two binomials. It stands for First, Outer, Inner, Last — the four pairs of terms you must multiply together. After computing all four products, you combine any like terms to get the simplified polynomial. For example, (a + b)(c + d) = ac + ad + bc + bd.

What does FOIL stand for?

FOIL stands for First, Outer, Inner, Last. 'First' means multiply the first term in each binomial together. 'Outer' means multiply the outermost terms. 'Inner' means multiply the two innermost terms. 'Last' means multiply the last term in each binomial. Combining all four results gives the expanded expression.

What is the distributive property and how does it relate to FOIL?

The distributive property states that a(b + c) = ab + ac. The FOIL method is simply a structured application of the distributive property applied twice — once for each term in the first binomial being distributed across the second binomial. FOIL ensures you never miss a term when expanding.

Can FOIL be used for expressions with subtraction?

Yes. Simply treat subtraction as adding a negative number. For example, (2x − 3)(x + 5) can be written as (2x + (−3))(x + 5). Enter −3 as the constant of the first binomial in this calculator. The signs are handled automatically in all four FOIL multiplications.

What does the result look like for two binomials?

Multiplying two first-degree binomials (ax + b)(cx + d) always produces a trinomial of the form Ax² + Bx + C, where A = ac (First), B = ad + bc (Outer + Inner combined), and C = bd (Last). This calculator displays all three coefficients clearly.

What if the coefficients cancel out and the result is a binomial?

If the Outer and Inner terms are equal and opposite (e.g., (x + 5)(x − 5)), the middle x terms cancel out, leaving a binomial of the form x² − 25. This is called the difference of two squares. The calculator will show a zero middle coefficient in that case.

Is FOIL the only way to multiply binomials?

No. You can also use the Box (area) method or simply apply the distributive property step by step. All methods produce the same result. FOIL is popular because it provides a memorable four-step checklist that prevents missing any term during manual calculation.

Can this calculator handle decimals or fractions as coefficients?

Yes. You can enter decimal values such as 0.5 or 1.25 as coefficients or constants. The FOIL method works identically for decimal inputs — all four term products are computed and combined just as with integers.

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