If you write a polynomial as the product of two or more polynomials, you have factored the polynomial. Here is an example:
The polynomials x-3 and are called factors of the polynomial . Note that the degrees of the factors, 1 and 2, respectively, add up to the degree 3 of the polynomial we started with. Thus factoring breaks up a complicated polynomial into easier, lower degree pieces.
We are not completely done; we can do better: we can factor
We have now factored the polynomial into three linear (=degree 1) polynomials. Linear polynomials are the easiest polynomials. We can't do any better. Whenever we cannot factor any further, we say we have factored the polynomial completely.