# The Unapologetic Mathematician

## 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

$p(x)=(X-x)q(x)=0$

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

$p=c_0+c_1X+c_2X^2+...+c_nX^n$

and let’s also say we’ve got a root $x$ so that $p(x)=0$. But that means

$0=c_0+c_1x+c_2x^2+...+c_nx^n$

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

$\begin{aligned}p=\left(c_0+c_1X+c_2X^2+...+c_nX^n\right)-\left(c_0+c_1x+c_2x^2+...+c_nx^n\right)\\=c_1(X-x)+c_2(X^2-x^2)+...+c_n(X^n-x^n)\end{aligned}$

Now for any $k$ we can use the identity

$X^k-x^k=(X-x)(X^{k-1}+X^{k-2}x+...+Xx^{k-2}+x^{k-1})$

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

## About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.
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