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.

differentiationon polynomials. This is no simple operation like this on integers, although people do think about getting good analogues of it (e.g., Buium). For example, a polynomial in Q[x] is squarefree iff it is relatively prime to its derivative, and gcd(f,f') can be computed rather efficiently. There is no simple way to determine if an integer is squarefree without in some way factoring it. – KConrad Oct 29 '10 at 6:20