Special Products

Three patterns come up constantly.

Square of a sum: (a + b)² = a² + 2ab + b²
Square of a difference: (a − b)² = a² − 2ab + b²
Difference of squares: (a + b)(a − b) = a² − b²

The middle term in the first two is where marks are lost. (x + 5)² is NOT x² + 25. It is x² + 10x + 25, because the outer and inner products both give 5x.

The third pattern has no middle term at all, because the outer and inner products cancel exactly.

Check the first one numerically if you doubt it: (2 + 3)² = 25, while 2² + 3² = 13.

Each pattern is an expansion whose middle behaviour follows from FOIL. In (a + b)² the outer and inner products are both ab, giving 2ab. In (a + b)(a − b) they are −ab and +ab, which cancel, leaving the difference of squares.

Recognising these in reverse is the fastest factoring available: a trinomial is a perfect square exactly when its first and last terms are squares and the middle equals twice the product of their roots.

The failure of (a + b)² = a² + b² is the single most persistent algebra error, and it recurs in every disguise: √(a + b) ≠ √a + √b, 1/(a+b) ≠ 1/a + 1/b, and sin(a + b) ≠ sin a + sin b. The general principle is that nonlinear operations do not distribute over addition.

Difference of squares generalises to a^n − b^n factorisations and underlies the conjugate technique for rationalising denominators, since (√a + b)(√a − b) = a − b² is radical-free.