Factoring a number involves breaking it down into all combinations of numbers that, when multiplied, result in the number. Most importantly, prime factoring is breaking down a number into any prime numbers that can be multiplied to get the number. Prime factoring is often a very useful way to factor numbers.
The distributive property of algebraic expressions that states that "a * (b + c) = ab + ac." Using the distributive property to factor algebraic expressions can lead the way to simplifying the problem by later dividing out common factors. The distributive property is also the cornerstone of more advanced factoring techniques.
This is a common factoring technique worth memorizing. When presented with an expression that is the difference between two perfect squares, i.e. "a^2-b^2," it should be immediately recognized that this factors as "(a-b)(a+b)." This is essentially an advanced use of the distributive property, and can be a useful technique for simplifying many kinds of algebraic expressions.
Quadratic equations are fairly common in algebra; they are defined as polynomials of the second degree. This means that one variable term is raised to the power of two. An example would be "x^2 + 4x + 4." Factoring these is accomplished by simpler factoring techniques, intuition and trial and error. The first term is split into constituent factors using the distributive property; in this example, it gives "(x + )(x + )." From the factors of the last term of the quadratic, 4, then find two such terms that, when added, sum to the middle term in the quadratic. In the example, the answer is "(x + 2)(x + 2)."