Reverse FOIL

"Reverse FOIL" is a method of factoring a quadratic trinomial by trial-and-error.  The strategy is to determine the First and Last terms of each binomial in the factored product so the Outer and Inner products add to the middle term.

The following examples illustrate this.

Example 4:  Factor  3x2 + 5x – 12

Solution:  First we concentrate on 3x2 and –12.  3x2 is the product of 3x and x, so we place these First terms of the factors:

3x2 + 5x – 12 = (3x           )(x          )

Now –12 could result in many different ways.  It could be –1 times 12, –2 times 6, 2 times –6, or any number of other products.

To make it easier, we ignore the negative for the time being and try various products that give 12, and then ask ourselves if it is possible to get the middle term, 5x, by subtracting the Outer and Inner products:

3x2 + 5x – 12 = (3x           2)(x          6)

                          Outer = 18x, Inner = 2x  No

3x2 + 5x – 12 = (3x           6)(x          2)

                          Outer = 6x, Inner = 6x  No

3x2 + 5x – 12 = (3x           4)(x          3)

                          Outer = 9x, Inner = 4x  Yes, 9x – 4x = 5x

So we want the 4x to be negative.  Therefore the factorization is:

3x2 + 5x – 12 = (3x – 4)(x + 3)


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