When we consider a polynomial as a function, we’re particularly interested in those field elements so that . We call such an a “zero” or a “root” of the polynomial .
One easy way to get this to happen is for to have a factor of . Indeed, in that case if we write for some other polynomial then we evaluate to find
The interesting thing is that this is the only way for a root to occur, other than to have the zero polynomial. Let’s say we have the polynomial
and let’s also say we’ve got a root so that . But that means
This is not just a field element — it’s the zero polynomial! So we can subtract it from to find
Now for any we can use the identity
to factor out from each term above. This gives the factorization we were looking for.