For example:

f(x)=x^2-4x+3

You want to solve: x^2-4x+3<0

You say: x^2-4x+3=0 <-> (x-1)(x-3)=0

You have solved it for f(x)=0 but you want to know the numbers where f(x)<0 so you draw a picture of the x-axis and mark the point where f(x) becomes zero so:

x---------1-------3--------

f(x) -----0-------0--------

You can now determine where f(x)<0 or f(x)>0 so a polynomial can change sings only at its zeros.