Reverse FOIL Calculator

Factor a quadratic trinomial into two binomials and see how the coefficients fit together.

Factor a trinomial
Enter the integer coefficients from ax² + bx + c.

About reverse FOIL and trinomial factoring

Reverse FOIL is a practical name for undoing the multiplication of two binomials. FOIL stands for first, outer, inner, and last, the four products produced when expressions such as (px + q)(rx + s) are multiplied. Combining those products gives prx² + (ps + qr)x + qs. A reverse FOIL calculator starts with the coefficients a, b, and c in ax² + bx + c and searches for integers p, q, r, and s that satisfy pr = a, qs = c, and ps + qr = b. For a monic trinomial, where a equals 1, the process is especially familiar. To factor x² + 5x + 6, find two numbers whose product is 6 and whose sum is 5. Those numbers are 2 and 3, so the factors are (x + 2)(x + 3). When the leading coefficient is not 1, both coefficients of x in the binomials must also be considered. For 2x² - 3x - 2, the factor pair (x - 2)(2x + 1) works because the outer and inner terms add to -3x. This calculator searches exact integer factor pairs, including negative factors and zero constants. It is designed for trinomials that factor cleanly over the integers. If no integer arrangement produces the middle coefficient, the tool reports that the expression is not factorable over the integers. That does not mean the quadratic has no roots. It may have irrational or complex roots that are better found with the quadratic formula. Factoring is useful because a product reveals a quadratic equation's zeros immediately. If (x + 2)(x + 3) equals zero, the zero-product property says x is -2 or -3. Factored form also helps simplify rational expressions, sketch parabolas, solve polynomial inequalities, and verify algebraic work. Always check a result by multiplying the binomials: the leading terms must reproduce ax², the last terms must reproduce c, and the two cross-products must combine to bx. Use integer inputs that preserve the original expression. A negative sign belongs in the coefficient field, while a missing middle term should be entered as zero. Common numerical factors do not need to be removed first because the search can still find a valid pair of binomials. However, recognizing a greatest common factor before reverse FOIL often makes hand calculations faster and produces a cleaner explanation.

Reverse FOIL examples

These trinomials show positive, negative, monic, and missing-middle-term cases.

TrinomialFactored formWhy it works
x² + 5x + 6(x + 2)(x + 3)2 × 3 = 6 and 2 + 3 = 5.
2x² - 3x - 2(x - 2)(2x + 1)The cross-products x and -4x combine to -3x.
x² - 9(x - 3)(x + 3)This is a difference of squares with a zero middle coefficient.
3x² - 2x - 5(x + 1)(3x - 5)The cross-products 3x and -5x combine to -2x.

How to use the reverse FOIL calculator

  1. Read the coefficient of x² and enter it as coefficient a.
  2. Enter the coefficient of x as b, using zero when the middle term is absent.
  3. Enter the signed constant term as c.
  4. Select Calculate Factors to search the integer factor pairs.
  5. Multiply the displayed binomials to verify that they reproduce the original trinomial.

Reverse FOIL calculator FAQ

What does reverse FOIL mean?

Reverse FOIL means turning a quadratic trinomial back into a product of two binomials. It reverses the first, outer, inner, and last multiplication process.

Can every trinomial be factored?

Not every trinomial factors using integer coefficients. Some require irrational or complex values, in which case the quadratic formula is a more suitable method.

How are negative constants handled?

A negative constant means the two constant factors have opposite signs. The sign arrangement must also make the cross-products add to the middle coefficient.

What if the middle term is missing?

Enter zero for coefficient b. The calculator will look for cross-products that cancel, as in a difference of squares.

How can I check the factored result?

Multiply the displayed binomials using FOIL and combine like terms. The resulting coefficients should exactly match a, b, and c from the original trinomial.