# Factorization

(mathematical) decomposition into a product

Factorization (also called factorisation and factoring) is taking a composite number apart into numbers that multiply together to get the original number. These smaller numbers are called factors or divisors. 1 is a factor of all numbers.

Prime factorization is breaking apart a composite numbers into prime numbers that can be multiplied to give the larger number. Note that since 1 is not prime, it is not included in the prime factorization.

For example, 12 can be factored as 4 × 3. Since 4 is not a prime number, that is not its prime factorization. 12's prime factorization is in fact 3 × 2 × 2.

The numbers which are obtained from the factorization are usually ordered, for example, starting with the smallest number. For example, 72=2^3*3^2. The factorization of every number is unique. This generalizes to:

1. Every number has a unique prime factorization
2. Every prime factorization corresponds to a unique number

Since finding the numbers to multiply together is very difficult for large numbers, this fact can be used in cryptography.

## Polynomials

This is how one type of polynomial is factored.

$x^{2}+{\color {Green}9x}+20$

Find two numbers that add up to 9 and can be multiplied to get 20. Here, these numbers are 4 and 5.

$=x^{2}+{\color {Green}4x+5x}+20$

$=(x^{2}+4x)+(5x+20)$

$=x(x+4)+5(x+4)$

$=(x+5)(x+4)$