Reverse FOIL Calculator
Factor an integer-coefficient quadratic into two binomials and verify every First, Outer, Inner, and Last product.
Quadratic coefficients
Required nonzero integer, from – 1,000,000 to 1,000,000.
Required integer. Enter the sign explicitly when negative.
Required integer. Zero is allowed and produces a factor containing x.
Accepted format: whole numbers only, with an optional leading + or – sign. Decimal points, grouping commas, fractions, and scientific notation are rejected rather than reinterpreted.
Live factorization
Factored form
(2x + 1)(3x – 5)
Multiplying the two binomials reproduces the original quadratic exactly.
Original polynomial
6x² – 7x – 5
First product
2 × 3 = 6
Outer product
2 × – 5 = – 10
Inner product
1 × 3 = 3
Middle coefficient check
– 10 + 3 = – 7
Last product
1 × – 5 = – 5
FOIL verification table
| Step | Multiplication | Coefficient | Expanded term |
|---|---|---|---|
| First | (2x)(3x) | 6 | 6x² |
| Outer | (2x)( – 5) | – 10 | – 10x |
| Inner | (1)(3x) | 3 | 3x |
| Last | (1)( – 5) | – 5 | – 5 |
The Outer and Inner coefficients combine to – 7, while the First and Last products reproduce 6 and – 5.
How to use this reverse FOIL calculator
What this calculator does. This tool takes the three integer coefficients in a quadratic expression of the form ax² + bx + c and searches for two integer-coefficient binomials whose product is exactly that expression. It then shows the First, Outer, Inner, and Last multiplications so you can verify the factorization rather than accepting an unexplained answer. It is designed for factoring over the integers. It does not claim that a polynomial has no real or complex roots when no integer-binomial factorization exists; it only reports that reverse FOIL does not produce integer factors.
When to use it. Use the calculator to check algebra homework, confirm a trial-and-error factorization, demonstrate why the signs in two binomials matter, or identify a quadratic that is irreducible over the integers. It is also useful before solving a quadratic equation by the zero-product property, because a correct factorization must expand back to the original coefficients. For a structured review of this method, see OpenStax's explanation of factoring trinomials of the form ax² + bx + c.
How to calculate. The calculator opens with the complete demonstration 6x² – 7x – 5, so the results and a validated example Excel workbook are available immediately.
- Replace the demonstration values in First coefficient, Second coefficient, and Third coefficient with the integers from your quadratic.
- Read Factored form. When factors are found, verify the four product cards and the FOIL verification table. When no factors are found, the result states that the expression is irreducible over the integers.
- Select Download Excel to export the current coefficients, factorization status, and verification rows as a real XLSX workbook. Select Reset to clear the demonstration and all calculated content. Reset may disable Download Excel until all three required fields contain a complete valid quadratic again.
Input guide. First coefficient is the required nonzero integer a multiplying x². Enter a whole number from – 1,000,000 to 1,000,000; for example, 6. A larger absolute value changes the possible first-term factor pairs. Entering 0 is invalid because the expression would no longer be quadratic. Second coefficient is the required integer b multiplying x; for example, – 7. Its sign and size determine whether the Outer and Inner products can add to the needed middle term. A common mistake is omitting the minus sign. Third coefficient is the required constant c; for example, – 5. Its factor pairs become the last terms of the binomials. Zero is valid and yields a factor containing x. All three fields accept only ungrouped whole-number notation with an optional leading sign; “1,000”, “2.5”, and “1e3” are intentionally rejected.
Output guide. The Polynomial pill and Original polynomial card restate the normalized expression. Status reports whether integer factors were found. Factored form is the product of the two binomials, or an irreducibility message. First product checks the x² coefficient. Outer product and Inner product are the two x-term contributions; their sum appears in Middle check and Middle coefficient check. Last product checks the constant. The FOIL verification table lists the same four products as First, Outer, Inner, and Last rows. These are exact algebraic identities for valid integer inputs, not statistical estimates.
Worked example. With a = 6, b = – 7, and c = – 5, the calculator selects 2 and 3 for the first terms because 2 × 3 = 6. It selects 1 and – 5 for the last terms because 1 × – 5 = – 5. The Outer product is 2 × – 5 = – 10 and the Inner product is 1 × 3 = 3. Their sum is – 10 + 3 = – 7, matching b. Therefore, 6x² – 7x – 5 = (2x + 1)(3x – 5). Expanding those binomials returns 6x² – 10x + 3x – 5, which simplifies to the original expression.
Learn more. The trial-and-error logic is also illustrated in West Texas A&M University's factoring-polynomials tutorial, while LibreTexts provides a direct reverse FOIL walkthrough.
How the integer factor search works
The calculator looks for integers α, β, γ, and δ that satisfy three conditions at once:
The first and last equations generate possible factor pairs. The middle equation is the decisive check: the Outer and Inner coefficients must combine to the exact second coefficient. The search is exhaustive within the supported integer range, so a “no integer-binomial factorization” result means no tested integer combination satisfies all three identities. The polynomial may still factor over rational, real, or complex numbers, or it may be solved by another method such as the quadratic formula.
Interpretation and common mistakes
- Keep the sign of every coefficient. A negative constant requires opposite signs in the last terms, while a positive constant uses matching signs.
- Do not compare only the first and last products. Many candidate pairs reproduce a and c but fail because the Outer and Inner sum does not equal b.
- If one coefficient is zero, the displayed polynomial may be a binomial rather than a trinomial. The calculator still factors it when an integer factorization exists.
- Equivalent answers can differ by order or by multiplying both binomials by – 1. The calculator presents one normalized representative and verifies it by expansion.