← Chapter 6
College Algebra · Chapter 6

Patterns You Can Factor on Sight

Four shapes show up so often that it is worth memorizing them — the difference of squares, the perfect square trinomial, and the sum and difference of cubes. Pick an example, walk it one step at a time, and watch the picture on the left prove why the rule is true.

Why the pattern works

Test it with real numbers

a7
b3

Know these on sight

perfect squares1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 perfect cubes1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 A variable power is a square when its exponent is even — halve it: √(x10) = x5. It is a cube when the exponent divides by 3 — take a third: the cube root of x12 is x4. 64 is both (82 and 43), so x6 − 64 can be opened either way. Start with the difference of squares — it factors further.
Four special patternspick an example to begin
SOAP keeps the cube signs straight. Reading (a ± b)(a2 ∓ ab + b2) left to right: Same sign as the original, then the Opposite sign, then Always Positive.
The trap: a sum of squares does not factor over the real numbers. x2 + 9, 4x2 + 25 and x2 + 1 are prime — leave them alone. Cubes are different: both a3 + b3 and a3 − b3 factor. And always pull out a common factor first: 2x2 − 18 is not a difference of squares until you write it as 2(x2 − 9).