FOIL can be used to factor quadratic expressions, that is, formulas of the type

ax^{2} + bx + c

I feel here focus on the simpler type where the first coefficient is 1, that is

x^{2} + bx + c

This can be factored into the product (x + d)(x + e)

by realizing that c = d*e (the "L" part of FOIL is just a number, no x)

and b = d + e (the "O" and "I" parts of FOIL together give dx + ex, that is (d+e)x)

(the "F" part is this special case is just x*x = x^{2})

This because of FOIL (try it out by multiplying out (x + d)(x + e).

So, as an example, let's say you need to factor x^{2} + 3x + 2, you know because of FOIL that the product has to be of the shape

(x + d)(x + e)

where d*e = 2

and d+e = 3

Can you think of two numbers which multiplied together give 2 and added together give 3?

Yes, 1 and 2! So the result of the factoring is

(x + 1)(x+2) or (x + 2)(x+1)

The order makes no difference.

Another example:

Factor x^{2} - 5x + 6

Steps: Find two numbers that together multiply to 6 and add to -5.

1. The middle term is negative, that means that at least one of the numbers is negative.

2. The last term is positive, that means that the two numbers have the same sign.

3. Together that means that both numbers are negative.

4. Find two negative numbers which multiply to 6: either -1 and -6 or -2 and -3.

5. Which pair of numbers adds to -5? -2 and -3.

Result: (x - 2)(x - 3)

If you want more examples, or want to know how it works for equations in which x^{2} has a coefficient other than 1, email me.