Why are polynomials easier to handle with than integers? This may seems to be an elementary question, but I found no answers on MO nor google.
I have always heard "polynomials are easier to handle with than integers". For example:


*

*When $n$ is quite large, maybe 200 or more, it's relatively easier to factorize a polynomial $f$ of degeree $n$ than to factorize an integer with $n$ bytes.

*When multiplying large integers, we see them as polynomials,use techniques such as FFT,intepolations to multiply polynomials,and then back to integers.
3.The zeta functions of $F[x]$ and $\mathbb{Z}$, and the former are easier to study than the latter.
Of course there are other examples, but because of my shortage of knowledge, I can only lise these above.
So my question is (as in the titile): Why are polynomials easier to handle with than integers? I ask this because contrary to our intuitives, polynomials are "more complex" objects than integers. 
 A: In the Euclidean division of polynomials, quotient and remainder are uniquely defined, and compatible with addition. This fails for integers. 
There is no integer analogue of constant polynomials. I guess that the search for the "field with one element" is in a sense motivated by this.
A: Divisibility of polynomials is a much more rigid property than in integers. Given an integer $n$, another $N$ has a non-zero probability $\frac1n$ to be a multiple of $n$. On the contrary, the probability (no rigour here) that a ``random'' polynomial $P\in{\mathbb C}[X]$ be a multiple of a given one $q$ is zero. For instance, $P$ is a multiple of $X$ iff $P(0)=0$, an event of probability $0$.
If you increase the structure, by either considering multi-variate polynomials, or polynomials with coefficients in $\mathbb Z$, you ``simplify'' even more, in the sense that you have additional criteria for divisibility of primality (Newton's polygon, Eisenstein's criterion, ...), and you have a huge theory (Galois' theory) which you can use and abuse.
Edit. An example of the powerness of polynomials that I like a lot is the following.

Theorem. Let $k$ be a field and $A_0,A_1,B_0,B_1\in M_n(k)$ be given, with $A_0,A_1$ invertible. Let $X$ be an indeterminate. If $XA_0+B_0$ and $XA_1+B_1$ are equivalent in $M_n(k[X])$, the there exist $G,H\in GL_n(k)$ such that $GA_0=A_1H$ and $GB_0=B_1H$.

The case where $A_0=A_1=I_n$ is at the basis of the theory of similarity invariants.
A: How Halloweeny can you get with your questions? The norm on $Z$ is Archimedean and the norm on $F[X]$ is non-Archimedean, and, in general, non-Archimedean maths is easier than Archimedean...
A: Following the commentary by Amri, I think that his idea can be explained in terms of graduations: The natural graduation of $F[X]$ over $F$, generated by the degree, allows us to compute simultaneously a lot of $F$-sums without mixing information. Maybe the rest of facts can be also described by graduation properties? (I'm not taking into account trivial graduations for $Z$).
