factor

factor, in mathematics, a number or algebraic expression that divides another number or expression evenly—meaning there is no remainder. For example, 3 and 6 are factors of 12 because 12 divided by 3 equals 4 and 12 divided by 6 equals 2, with no remainder.

A prime number (or prime algebraic expression) is one that has only two factors: itself and 1. A number or expression that has more than two factors is called composite. The prime factors of a number or expression are simply the factors that are prime. According to the fundamental theorem of arithmetic, every whole number greater than 1 can be written as a product of its prime factors in a way that is unique, except for the order of the factors. For instance, the number 60 can be written as the product 2 × 2 × 3 × 5.

Factoring—breaking a number or expression down into its factors—is important in basic arithmetic and its advanced applications. For example, methods for factoring large whole numbers play a key role in public-key cryptography, which is used to secure information on the Internet. Factoring is also a particularly important step in the solution of many algebraic problems. For example, consider the polynomial equation$x^2 - x - 2 = 0$.This equation can be rewritten (or factored) as$\hspace{1em}(x - 2)(x + 1) = 0$.In algebra if the product of two expressions is zero, then at least one of the expressions must be zero. So we solve$x - 2 = 0$ and $x + 1 = 0$which gives the solutions$x = 2$ and $x = -1$.