You first came across distributing when you were learning how to clean up equations and isolate a variable. Essentially, distributing is applying multiplication across a sum.
To review, if you are presented with the expression 3(x + 2), and you want to simplify it, you have to distribute the 3 so that it is multiplied by both the x and the 2:
But what if the first part of the multiplication is more complicated? Suppose you need to simplify (a + b)(x + y)?
Simplifying this expression is really an extension of the principle of distribution—every term in the first part of the expression must multiply every term in the second part of the expression. In order to do so correctly every time, you can use a handy acronym to remember the steps necessary: FOIL. The letters stand for First, Outer, Inner, Last.
In this case, it looks like this:
| (a + b)(x + y) | F − multiply the first term in each of the parentheses: a × x = ax. |
| (a + b)(x + y) | O – multiply the outer term in each: a × y = ay. |
| (a + b)(x + y) | I – multiply the inner term in each: b × x = bx. |
| (a + b)(x + y) | L – multiply the last term in each: b × y = by. |
So you have (a + b)(x + y) = ax + ay + bx + by.
You can verify this system with numbers. Take the expression (3 + 4)(10 + 20). This is no different than multiplying (7)(30), which gives you 210. See what happens when you FOIL the numbers:
| (3 + 4)(10 + 20) | F − multiply the first term in each of the parentheses: 3 × 10 = 30. |
| (3 + 4)(10 + 20) | O – multiply the outer term in each: 3 × 20 = 60. |
| (3 + 4)(10 + 20) | I – multiply the inner term in each: 4 × 10 = 40. |
| (3 + 4)(10 + 20) | L – multiply the last term in each: 4 × 20 = 80. |
Finally, sum the four products: 30 + 60 + 40 + 80 = 210.
Now that you have the basics down, go through a more GRE-like situation. Take the expression (x + 2)(x + 3). Once again, begin by FOILing it:
| (x + 2)(x + 3) | F − multiply the first term in each of the parentheses: x × x = x2. |
| (x + 2)(x + 3) | O – multiply the outer term in each: x × 3 = 3x. |
| (x + 2)(x + 3) | I – multiply the inner term in each: 2 × x = 2x. |
| (x + 2)(x + 3) | L – multiply the last term in each: 2 × 3 = 6. |
The expression becomes x2 + 3x + 2x + 6. Combine like terms, and you are left with x2 + 5x + 6. The next section will discuss the connection between distributing, factoring, and solving quadratic equations. But for the moment, practice FOILing expressions.
FOIL the following expressions.
(x + 4)(x + 9)
(y + 3)(y − 6)
(x + 7)(3 + x)