Three quadratic expressions called special products come up so frequently on the GRE that it pays to memorize them. You should immediately recognize these three expressions and know how to factor (or distribute) each one automatically. This will usually put you on the path toward the solution to the problem:
| Special Product #1: | x2 − y2 = (x + y)(x − y) |
| Special Product #2: | x2 + 2xy + y2 = (x + y)(x + y) = (x + y)2 |
| Special Product #3: | x2 − 2xy + y2 = (x − y)(x − y) = (x − y)2 |
You should be able to identify these products when they are presented in disguised form. For example, a2 − 1 can be factored as (a + 1)(a − 1). Similarly, (a + 1)2 can be distributed as a2 + 2a + 1.
Avoid the following common mistakes with special products:
| Wrong: | (x + y)2 = x2 + y2 ? | Right: | (x + y)2 = x2 + 2xy + y2 |
| (x − y)2 = x2 − y2 ? | (x − y)2 = x2 − 2xy + y2 |
Factor the following.
4a2 + 4ab + b2 = 0
x2 + 22xy + 121y2 = 0