How is the factor theorem used?

The factor theorem is used to determine factors of large polynomials so that we can split large polynomials into a product of linear polynomials. Say we have a cubic polynomial of the form f(x)=x^3+bx^2+cx+d and we want to know if (x-a) is a factor we need only work out the value of f(a). We have that (x-a) is a factor if and only if f(a)=0 and so if f(a) is not equal to 0 then (x-a) is not a factor. If we want to know if (x+a) is a factor we simply find the value of f(-a).
We can understand the example above by factorising f(x). If we assume (x-a) is a factor of f(x) then we can write f(x)=(x-a)(x^2+ex+f). Here we can see that f(a)=(a-a)((x^2+ex+f) and so f(a)=0.