What is Reverse FOIL?
FOIL stands for First, Outer, Inner, Last -- a method for multiplying two binomials. Reverse FOIL is the process of factoring a trinomial back into two binomials. Given a trinomial ax² + bx + c, the goal is to find binomials (px + q)(rx + s) such that when multiplied using FOIL, they produce the original trinomial.
Factoring Methods
Simple Case (a = 1)
Find two numbers that multiply to c and add to b.
x² + bx + c = (x + m)(x + n) where m*n = c, m+n = b
AC Method (a != 1)
Find two numbers that multiply to a*c and add to b, then factor by grouping.
Find m, n: m*n = a*c and m+n = b
Quadratic Formula
When factoring by inspection is difficult, use the quadratic formula to find roots.
x = (-b +/- sqrt(b²-4ac)) / 2a
When Can a Trinomial Be Factored?
- The discriminant (b² - 4ac) must be non-negative for real factors.
- For integer factoring, the discriminant must be a perfect square.
- If the discriminant is negative, the trinomial has no real factors (only complex).
Examples
- x² + 5x + 6 = (x + 2)(x + 3) -- factors of 6 that add to 5 are 2 and 3
- 2x² + 7x + 3 = (2x + 1)(x + 3) -- AC method with a*c = 6
- x² - 4 = (x + 2)(x - 2) -- difference of squares
- x² + 1 -- cannot be factored over the reals