FOIL Sucks...

FOIL is a conceptually free algorithm, at least as it is retained by students. It's effectiveness is limited to multiplying binomials with integer coefficients. Anything more complex is a new "concept." Further, the reverse, factoring, especially by grouping, is extremely difficult to understand without first understanding the distributive property (where the FOIL method originates).

Example:
(a + 2)(b - 3) = ab - 3a + 2b - 6 by the FOIL method.

(a + 2)(b - 3) = a(b - 3) + 2(b - 3)= ab - 3a + 2b - 6

By the distributive property (which states that everything in the first parenthesis multiplied by all in the other).

So, in this example, FOIL is faster, if you write everything out.

Example:
(a + 2)(b + c -3) = ...with FOIL, nada, zip.

(a + 2)(b + c - 3)

With distributive property ...nothing new, everything in the first parenthesis multiplied by the second.

(a + 2)(b + c - 3) = a(b + c - 3) + 2(b + c - 3) = ab + ac - 3a + 2b + 2c - 6

Example: Factor x^3 - 2x^2 -9x + 18

Looking at this as the opposite of distributing is easy:

Factoring

x^3 - 2x^2 -9x + 18
(x^3 - 2x^2) (-9x + 18)
x^2(x - 2) - 9(x - 2)
(x^2 - 9)(x- 2)

Last example:

(sqrt(x - 1) + 1)(sqrt(x -1) + 1)
FOIL...even college algebra students are stuck. Distributive property, easy.