special products

Special products are a handful of multiplication patterns that come up so often it pays to recognize them on sight, instead of grinding through FOIL each time. They are shortcuts that turn a multiplication into a quick fill-in-the-blanks.

The headline three are: the square of a sum, (a + b)^2 = a^2 + 2ab + b^2; the square of a difference, (a - b)^2 = a^2 - 2ab + b^2; and the difference of two squares, (a + b)(a - b) = a^2 - b^2. Notice that in the last one the middle terms cancel, leaving only two terms.

These same patterns run in reverse for factoring: spotting a^2 - b^2 lets you write (a + b)(a - b) instantly, and recognizing a perfect-square trinomial lets you collapse it to (a + b)^2. The one error to avoid is the classic (a + b)^2 = a^2 + b^2 — the middle term 2ab is real and must never be dropped.