Reverse FOIL
Also known as: factoring trinomials
Reverse FOIL is a method for factoring a quadratic trinomial back into two binomials. It undoes the FOIL multiplication pattern by finding two numbers that multiply to the constant term and add to the middle coefficient.
Reverse FOIL is factoring: taking a quadratic like x² + 5x + 6 and rewriting it as the product of two binomials, (x + 2)(x + 3). The name comes from FOIL — First, Outer, Inner, Last — the pattern used to multiply two binomials. Reverse FOIL runs that process backward, recovering the binomials from the expanded trinomial.
For a trinomial of the form x² + bx + c, the method is a number hunt: find two numbers that multiply to c and add to b. For x² + 5x + 6, the numbers 2 and 3 multiply to 6 and add to 5, so the factors are (x + 2)(x + 3). Signs carry information: in x² − 7x + 12, both numbers must be negative (−3 and −4), while in x² + x − 12, they have opposite signs (4 and −3).
Reverse FOIL matters because factoring is the fastest route to solving many quadratic equations. Once x² + 5x + 6 = 0 becomes (x + 2)(x + 3) = 0, the zero product property gives the solutions immediately: x = −2 or x = −3. Always check your work by FOILing the factors back out — you should land exactly on the original trinomial.
Factoring by reverse FOIL is tested across the major standardized tests. The GRE covers it in its FOIL and quadratic equations material, and the SAT and CLT both lean on it in their quadratics sections — typically asking you to factor, solve by factoring, or match a quadratic to its roots.
Key takeaways
- Reverse FOIL factors a quadratic trinomial into two binomials by undoing the FOIL multiplication pattern.
- For x² + bx + c, find two numbers that multiply to c and add to b.
- The signs of b and c tell you the signs of the two numbers before you start guessing.
- Factoring plus the zero product property is often the fastest way to solve a quadratic equation.
- The GRE, SAT, and CLT all test factoring quadratics by reverse FOIL.