Factorization

Factorization is a mathematical method for expressing a number or algebraic expression as a product of several components, called factors. For example, 12 can be written as 3 × 4 or 2 × 2 × 3. Similarly, the expression x² − 9 can be written as (x − 3)(x + 3).

Factorization is the inverse of multiplication. When factors are multiplied, they combine into a single larger expression; when an expression is factored, it is rewritten as a product. This method is used in solving equations, simplifying fractions, and finding common factors of numbers.

Factorization of numbers

Every integer greater than 1 has a unique factorization into prime numbers, except for the order of the factors. This principle is called the fundamental theorem of arithmetic. For example, 60 = 2 × 2 × 3 × 5. A prime number itself cannot be factored further except as 1 and itself.

Factorization of algebraic expressions

In algebra, there are standard methods for factorization. The most common include: extracting a common factor (for example 6x + 9 = 3(2x + 3)); difference of two squares (a² − b² = (a − b)(a + b)); and factorization of quadratic trinomials, such as x² + 5x + 6 = (x + 2)(x + 3). The last method depends on finding two numbers whose product gives the constant term and whose sum gives the coefficient of x.

Connection to roots

When a polynomial is factored into the form (x − r) multiplied by other factors, r is a root of the polynomial, meaning the expression equals zero when x = r. This is why factorization is a fundamental method for solving polynomial equations: if a product equals zero, at least one of its factors must be zero.

Applications

Factorization is used in school and higher education. It simplifies algebraic fractions, finds the least common multiple and greatest common divisor, and is applied in cryptography, where the difficulty of factoring large numbers is used for security purposes.