# Prime factors

A factor is a term in multiplication. A prime factor is a factor that is a prime quantity, meaning that it can only be formed as the product of 1 and itself. For example, 3 × 7 = 21. In this problem, 3 and 7 are prime factors, since they are both prime numbers.

Numbers are not the only quantities that can be prime factors. Expressions such as x, (x + 1), and (x^{2} - 5) are all also prime factors, since they cannot be factored further (can only be formed as the product of 1 and themselves).

## Prime factorization

The prime factorization of a given composite (not prime) quantity is the product of all its prime factors. There are a number of ways to determine the prime factorization of a given quantity, such as using a factor tree or trial division. Trial division is a straightforward method, but can be very tedious. It involves dividing a composite quantity (and the subsequent quotient) by the smallest prime numbers until the quotient is 1. The prime factorization is the product of all the prime numbers used in the division.

Example

Find the prime factorization of 60 using trial division.

64 ÷ 2 = 30

30 ÷ 2 = 15

15 ÷ 3 = 5

5 ÷ 5 = 1

In the example above, the prime factors are all underlined. The prime factorization of 60 is the product of all the prime factors:

60 = 2 × 2 × 3 × 5 = 2^{2} × 3 × 5

Therefore, the prime factors of 60 is 2^{2} × 3 × 5.

## Factoring

Factoring is the equivalent of prime factorization. The terms are essentially interchangeable, but "prime factorization" is conventionally used to refer to integers while "factoring" is used for algebraic expressions, such as polynomials. Factoring polynomials can be relatively simple or complicated depending on the polynomial. Below is a simpler example just to demonstrate the factorization of a polynomial in the context of prime factors.

Examples

1. Find the prime factors of 2x^{2} + 8.

2x^{2} + 8 = 2(x^{2} + 4)

Neither 2 nor x^{2} + 4 can be factored further, so they are the prime factors of 2x^{2} + 8.

2. Find the prime factors of x^{2} + 3x + 2.

x^{2} + 3x + 2 = (x + 2)(x + 1)

Neither x + 2 nor x + 1 can be factored further, so they are the prime factors of x^{2} + 3x + 2.