Square of a binomial
Special products are FOIL results worth memorizing. The first is the square of a binomial: (a + b)^2 = a^2 + 2ab + b^2. FOIL it once and you see why — the Outer and Inner are both ab, so they double. The middle term is twice the product of the two parts. A common error is writing (a + b)^2 = a^2 + b^2; that forgets the 2ab.
(x + 5)^2 = (x + 5)(x + 5) F: x·x = x^2 O: x·5 = 5x I: 5·x = 5x L: 5·5 = 25 = x^2 + 5x + 5x + 25 = x^2 + 10x + 25 ← 10x is 2·(x·5) With a minus: (x − 5)^2 = x^2 − 10x + 25
The result a^2 + 2ab + b^2 is exactly a perfect-square trinomial — a trinomial that came from squaring a binomial. Recognizing it later, in reverse, is the heart of factoring and of completing the square.
Difference of two squares
The second pattern: (a + b)(a − b) = a^2 − b^2. The same two parts, once added and once subtracted. FOIL it and the Outer (−ab) and Inner (+ab) cancel exactly, leaving only a^2 − b^2 — the difference of two squares. No middle term survives.
(x + 7)(x − 7) F: x·x = x^2 O: x·(−7)= −7x I: 7·x = +7x ← cancels the −7x L: 7·(−7)= −49 = x^2 − 49
Why bother memorizing
These patterns run both directions. Forward, they let you expand in one line instead of four. Backward, they let you spot a factorable form instantly: x^2 − 49 is a difference of squares, so it factors as (x + 7)(x − 7). That reverse reading is what makes factoring quick.