How to Reverse a FOIL When Factoring a Polynomial


The FOIL method is a mathematical key for reducing two binomial expressions to a trinomial mathematical expression by multiplying two sets of binomials by eachother. FOIL stands for first, outside, inside, and last, as this is the order of the multiplication that you would do to the variables in a two-binomial equation to reduce it. The reverse-FOIL method, then, is a process of doing FOIL backward. Reverse-FOIL is called for when you are trying to convert a trinomial back into a double binomial expression.

  • Set up two sets of parentheses under your trinomial equation like so: ( +/- )( +/- ). This is the form to which you will be converting the equation.

  • Factor the polynomial. For example, let's say our trinomial is 3x^2 + 10x + 8. To get 3x to the second power, we must multiply 3x by x. So, knowing this, we can fill in the first two variables in our parentheses: (3x +/- )( x +/- ).

  • Think of two factors of 8. Your possibilities are 1 and 8, 2 and 4 and their negatives.

  • Plug these number groups into the equation. For example, try the first possible factors of 1 and 8: (3x + 1)(x + 8). Now we check that with the FOIL method. The result we get is 3x^2 + 24x + x + 8, which is reduced to 3x^2 + 25x + 8. But this is not our original trinomial, so 1 and 8 are not the correct factors.

  • Keep plugging in the factors of 8 and checking it with FOIL. If you do, you will find that when you try 4 and 2: (3x + 4)(x + 2), this factors out to 3x^2 +10x +8. This is our original equation. We did it!


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