factoring

Factoring is the reverse of multiplying out. When you multiply, you start with pieces and combine them into one expanded expression; when you factor, you start with the expanded expression and break it back into the pieces — the factors — that were multiplied together. It is like taking the number 12 and writing it as 3 times 4, except now the things being multiplied are polynomials.

Concretely, to factor a polynomial means to write it as a product of two or more polynomials of lower degree (or of a number times such polynomials). For example, x^2 + 5x + 6 factors as (x + 2)(x + 3), because multiplying those two binomials back out recovers the original. Every factoring you do can be checked instantly by expanding the product and seeing whether you get back what you started with.

Factoring matters because a product is much easier to reason about than a sum. If you can rewrite an expression as a product, you can find where it equals zero (each factor gives a root), cancel common factors in a fraction, and reveal hidden structure. The honest caveat: not every polynomial factors nicely over the numbers you are allowed to use, and finding factors is often a search rather than a formula.