The Distributive Property and the FOIL Method
In an episode of Veronica Mars, a character describes the secret to passing Algebra: "F-O-I-L. That's all it is. First outside inside last. All Algebra, it's just the formula."† That's not actually true—most of Algebra has nothing to do with FOIL—but it does bring up a good point: what is the FOIL method anyway, and how do you use it?
Before we can talk about what the FOIL method is, we have to talk about the distributive property. This rule says that if you have multiple terms being added together in parentheses, then multiplied by something, then you can "distribute" the multiplication to each of the things being added, rather than dealing with the addition first. For example, if you have $4(7 + 19)$, the distributive property says that you can take the "multiply by 4" and apply it to the 7 and the 19 separately, turning it into $4·7 + 4·19$.
This is especially useful for situations where the things being added are not like terms and can't be added together, such as turning $(3x + 2y - 8)·9$ into $3x·9 + 2y·9 - 8·9$.
The distributive property can also be applied multiple times, in cases where you have multiple sets of parentheses being multiplied together. For example, if we want to simplify $(4x^2-8x +16)(7x-4)$, we can first distribute $(7x-4)$ to everything in $(4x^2-8x +16)$, which gives us $4x^2·(7x-4)-8x(7x-4) +16(7x-4)$. Then, we can distribute each term onto the $7x$ and the $-4$, so we get $4x^2·(7x)+4x^2·(-4)+(-8x)·(7x)+(-8x)·(-4) +16·(7x)+16·(-4)$, which can then be multiplied out and simplified.
The FOIL method, as it turns out, is a special case of using the distributive property. It refers specifically to situations where you're multiplying two sets of parentheses with two terms each, such as $(5x-7)(8y+4)$ or $(3a+2b)(9a-16b)$. For example, if we want to use the distributive property on $(5x-7)(8y+4)$, we can first distribute $(5x-7)$ to everything in $(8y+4)$, giving us $(5x-7)·8y + (5x-7)·4$, then distribute the $8y$ and the $4$ onto the $(5x-7)$, giving us $5x·8y - 7·8y + 5x·4- 7·4$, which can be multiplied out to $40xy - 56y + 20x - 28$.
Notice how distributing multiple times like this ends up having each term from the first group multiplied with each term from the second group. So instead of distributing one group, then the other group, as two separate steps, we could just skip directly to multiplying each term from the two groups. This is where FOIL comes into play!
Since it can be easy to lose track of which terms have been multiplied with which, "FOIL" is an acronym to remind us of all the different possible combinations. It stands for "First, Outside, Inside, Last" - we need to multiply together the first term in each group (5x and 8x), the "outside" terms for the whole expression (5x and 4), the "inside" terms (-7 and 8x), and the last term in each group (-7 and 4).
If we apply FOIL to $(3a+2b)(9a-16b)$, it looks like this: first, $3a·9a$ or $27a^2$, plus outside, $3a·(-16b)$ or $-48ab$, plus inside, $2b·9a$ or $18ab$, plus last, $2b·(-16b)$ or $-32b^2$. So we'd have $27a^2-48ab+18ab-32b^2$, and combining like terms would give us $27a^2-30ab-32b^2$ as the final answer.
Now it's your turn!
Use the FOIL method to simplify the following expressions:
$(-4x+13)(9x+12)$.
$(3x-6)(18x-5)$
Apply the distributive property to simplify the following expressions:
$3(12x + 11)$.
$-15(19x^2 - 10x - 2)$
$(2x+3y)(-17x +7y +6)$
