Factor

In mathematics, to factor a number or algebraic expression is to find parts whose product is the original number or expression. For instance, 12 can be factored into the product 6 × 2, or 3 × 4. The expression (x² - 4) can be factored into the product (x + 2)(x - 2). Factor is also the name given to the parts. We say that 2 and 6 are factors of 12, and (x-2) is a factor of (x² - 4). Thus we refer to the factors of a product and the product of factors.

The fundamental theorem of arithmetic states that every positive integer can be expressed as the product of prime factors in essentially a single way. A prime number is a number whose only factors are itself and 1 (the first few prime numbers are 1, 2, 3, 5, 7, 11, 13). Integers that are not prime are called composite. The number 99 is composite because it can be factored into the product 9 × 11. It can be factored further by noting that 9 is the product 3 × 3. Thus, 99 can be factored into the product 3 × 3 × 11, all of which are prime. By saying "in essentially one way," it is meant that although the factors of 99 could be arranged into 3 × 11 × 3 or 11 × 3 × 3, there is no factoring of 99 that includes any primes other than 3 used twice and 11.

Factoring large numbers was once mainly of interest to mathematicians, but today factoring is the basis of the security codes used by computers in military codes and in protecting financial transactions. High-powered computers can factor numbers with 50 digits, so these codes must be based on numbers with a hundred or more digits to keep the data secure.

In algebra, it is often useful to factor polynomial expressions (expressions of the type 9x³ + 3x² or x⁴ 27xy + 32). For example x² + 4x + 4 is a polynomial that can be factored into (x + 2)(x + 2). That this is true can be verified by multiplying the factors together. The degree of a polynomial is equal to the largest exponent that appears in it. Every polynomial of degree n has at most n polynomial factors (though some may contain complex numbers). For example, the third degree polynomial x³ + 6x² + 11x + 6 can be factored into (x + 3) (x² + 3x + 2), and the second factor can be factored again into (x + 2)(x + 1), so that the original polynomial has three factors. This is a form of (or corollary to) the fundamental theorem of algebra.

In general, factoring can be rather difficult. There are some special cases and helpful hints, though, that often make the job easier. For instance, a common factor in each term is immediately factorable; certain common situations occur often and one learns to recognize them, such as x³ + 3x² + xy = x(x²+ 3x + y). The difference of two squares is a good example: a² - b² = (a + b)(a - b). Another common pattern consists of perfect squares of binomial expressions, such as (x + b)². Any squared binomial has the form x² + 2bx + b². The important things to note are: (1) the coefficient of x² is always one (2) the coefficient of x in the middle term is always twice the square root of the last term. Thus x² + 10x + 25 = (x+5)², x² - 6x + 9 = (x-3)², and so on.

Many practical problems of interest involve polynomial equations. A polynomial equation of the form ax² + bx + c = 0 can be solved if the polynomial can be factored. For instance, the equation x² + x - 2 = 0 can be written (x + 2)(x - 1) = 0, by factoring the polynomial. Whenever the product of two numbers or expressions is zero, one or the other must be zero. Thus either x + 2 = 0 or x - 1 = 0, meaning that x = -2 and x = 1 are solutions of the equation.