The Unapologetic Mathematician

Mathematics for the interested outsider

Roots of Polynomials I

When we consider a polynomial as a function, we’re particularly interested in those field elements x so that p(x)=0. We call such an x a “zero” or a “root” of the polynomial p.

One easy way to get this to happen is for p to have a factor of X-x. Indeed, in that case if we write p=(X-x)q for some other polynomial q 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 x so that p(x)=0. But that means


This is not just a field element — it’s the zero polynomial! So we can subtract it from p to find


Now for any k we can use the identity


to factor out (X-x) from each term above. This gives the factorization we were looking for.

July 30, 2008 Posted by | Algebra, Polynomials, Ring theory | 2 Comments