factor (verb)

To factor is to ask “what could I have multiplied to get this?” It is the reverse of expanding: instead of multiplying things out into a sum, you write the expression as a product of simpler pieces, called factors.

Concretely, factoring rewrites a sum as a product. You might pull out a common factor — 6x + 9 = 3(2x + 3) — or recognize a pattern — x^2 - 9 = (x - 3)(x + 3). The product form is equivalent to the original (you could expand it back), but it exposes structure: the factors of a polynomial reveal where it equals zero.

Factoring is genuinely harder than expanding, because you must search for the pieces rather than just multiply given ones. Not every expression factors nicely over the integers — x^2 + 1, for instance, has no real factors. When that happens, the expression is called prime, and there is nothing wrong with leaving it as is.