Has the problem of factoring (over the rationals) the general trinomial $ax^n+bx^k+c$ with $a,b,c\in\mathbb{Z}$, $n,k\in\mathbb{N}, n>k>1$ been solved? By *solved* I mean a classification theorem which will either give the factors or give a certificate that it is irreducible.


One could ask the same question, but about finding *expressions* for the roots of a trinomial. I am aware that [long ago](http://library.wolfram.com/examples/quintic/timeline.html) this problem was worked on, but I have been unable to find a modern treatment of this problem.


The question which actually motivates this one is: given a trinomial which factors, consider the *number of terms* in each of the factors. Further, assume that the trinomial is **not** cyclotomic. How many terms can the factors have? [The cyclotomic case is conjectured to be special].


[Edit: clarified that I am interested in this special case, not the general case, where the algorithms of M. van Hoeij are the current best, much superior to Cantor-Zassenhaus].



Factoring and solving trinomials
One can of course apply general algorithms for irreducibility testing and factorization, so I presume you are asking if there is something more efficient or more explicit that can be said in the case of trinomials. Except for special cases I don't believe that is the case. 


While it is known that every binomial in $\mathbb{Q}(x)$ must have an irreducible factor that is either binomial or trinomial, no analogous bound is known in the trinomial case. It is at least 8 terms due to the [known [1]](http://journals.cambridge.org/action/displayFulltext?type=1&fid=5054152&jid=GMJ&volumeId=34&issueId=01&aid=5054144) example


$$f(x)f(-x) = - x^{14} - 27180501562500 x^2 + 1244325625000000$$


for $f(x) = x^7 + 20 x^6 + 200 x^5 + 2450 x^4 + 29000 x^3 + 545000 x^2 + 8101250 x + 35275000$


[1](http://journals.cambridge.org/action/displayFulltext?type=1&fid=5054152&jid=GMJ&volumeId=34&issueId=01&aid=5054144) Choudhry and A. Schinzel (1992).  

On the number of terms in the irreducible factors of a polynomial over $\mathbb Q$.  

Glasgow Mathematical Journal (1992), 34, 11-15.


To dig deeper I suggest starting with the work of Schinzel - who has studied these and related factorization problems intensively for almost half a century, e.g. see



> 
> MR1254093 (95d:11146) 11R09 (12E05 12E10)  
> 
>  Schinzel, Andrzej. 
>  On reducible trinomials.  
> 
>  Dissertationes Math. (Rozprawy Mat.) 329 (1993), 83 pp. 
> 
> 
> Let $K$ be a field. It is well known
>  that a binomial $x^n+a\in K[x]$ is
>  reducible iff it has the form
>  $x^{pk}-b^p$ ($p$ prime) or
>  $x^{4k}+4b^4$. In this treatise the
>  reducibility of trinomials
>  $x^n+ax^m+b$ $(a,b\neq 0)$ is
>  investigated. It turns out that the
>  situation is very complicated. A
>  satisfactory answer is obtained if $K$
>  is a rational function field. For
>  algebraic function fields in one
>  variable and for algebraic number
>  fields, less complete results are
>  proved. It is assumed throughout that
>  the characteristic of $K$ does not
>  divide $mn(n-m)$. 
> 
> 
> It is easy to find trinomials with
>  linear or quadratic factors. Table 1
>  of this paper provides additional
>  families of reducible trinomials if
>  $(n,m)$ belongs to a list of 12 pairs,
>  the largest being $(15,5)$. Perhaps
>  the simplest example is
>  $$x^6+4(v+1)x^2-v^2=(x^3+2x^2+2x-v)(x^3-2x^2+2x+v).$$ Every reducible trinomial
>  $f(x)=x^n+ax^m+b$ gives rise to
>  additional examples by considering
>  $u^nf(x^l/u)$ (with $u\in K^\times$
>  and $l\geq 1$) or $x^nf(1/x)/b$.
>  Theorem 1 essentially states that
>  every reducible trinomial arises in
>  this manner from the examples
>  indicated before if $K$ is a rational
>  function field. (More precisely, it is
>  assumed that $a^{-n}b^{n-m}$ is not a
>  constant.) Table 2 lists $7$ families
>  of reducible trinomials
>  $x^n+A(v,w)x^m+B(v,w)$ with $(v,w)\in
> > E(K)$, where $A$, $B$ are polynomials
>  over $\mathbb Z$ and $E$ is an
>  elliptic curve defined by an equation
>  $z^2=C(w)$, where $C$ is a monic
>  polynomial over $\mathbb Z$. The
>  polynomials $A,B$ and the
>  corresponding factorizations of the
>  trinomials are too complicated to be
>  included in this review. (For the
>  largest pair $(n,m)=(21,7)$ the
>  corresponding $A$ fills 10 lines in
>  the paper.) In Theorem 2 it is assumed
>  that $K$ is a finite extension of a
>  rational function field $F(t)$ such
>  that $\overline FK$ has genus $g>0$
>  and $a^{-n}b^{n-m}\notin
> > \overline{F}$. If $g=1$ then there are
>  no additional examples of reducible
>  trinomials. If $g>1$ then essentially
>  new examples with $n<24g$ may exist.
>  Theorem 3 reduces the case where $K$
>  is a finite separable extension of
>  $F(t)$ and $a^{-n}b^{n-m}\in\overline
> > F$ to studying reducibility over
>  $K\cap\overline F$. If $K$ is an
>  algebraic number field then for fixed
>  $n$, $m$ a finite number of
>  essentially new examples of reducible
>  trinomials $x^n+ax^m+b$ may exist
>  (Theorem 6). The author conjectures
>  that for every $K$ there is only a
>  finite number of these ``sporadic
>  trinomials''. If the conjecture holds
>  then there exists a constant $c(K)$
>  such that every trinomial over $K$ has
>  an irreducible factor with at most
>  $c(K)$ nonzero coefficients
>  (Consequence 2). Table 5 contains all
>  52 sporadic trinomials over $\mathbb
> > Q$ known to the author. Their degrees
>  lie in the range from $8$ to $52$. The
>  rest of the paper is devoted to
>  studying the reducibility of
>  $ax^n+bx^m+c\in\mathbb Z[x]$. Theorem
>  9 (refining a result of Nagell)
>  derives necessary conditions, which in
>  the case $(m,n)=1$ yield an explicit
>  bound for $b$ in terms of $a,c,m,n$.
>  For every positive integer $d$ there
>  exist only finitely many $n,m,b$ with
>  $n/(m,n)>d$ and $|b|>2$ such that
>  $x^n+bx^m\pm 1$ has a factor of degree
>  $d$; and these can be effectively
>  computed. Theorem 10 derives necessary
>  conditions from the existence of a
>  factor (of $ax^n+bx^m+c)$ of given
>  degree $d$. These imply that there
>  exists $n\_0(d)$ such that $x^n+bx^m+1$
>  is irreducible if $n\geq n\_0(d)$,
>  $n\neq 2m$, $|b|>2$. By Theorem 8, for
>  every $n$ there exist only finitely
>  many reducible trinomials $x^n+bx^m+1$
>  with $n\neq 2m$. 
> 
> 
> The proof of Theorem 10 does not
>  depend on the other results of the
>  paper. The same applies to Theorem 9.
>  All other theorems except for Theorem
>  3 are based on lower estimates for the
>  genus of certain function fields.
>  These estimates show that the
>  existence of a factor of degree $k$ of
>  $x^n+ax^m+b\in K[x]$ imposes severe
>  restrictions on $k,m,n,a,b$ provided
>  $K$ is a function field. The remaining
>  cases are treated in a long series of
>  lemmas applying to every field $K$
>  whose characteristic does not divide
>  $mn(n-m)$. In several cases the proofs
>  require extensive manipulations (with
>  polynomials in several variables)
>  which were performed by means of
>  computer algebra systems. Faltings'
>  theorem (solving Mordell's conjecture)
>  is invoked in the proof of Theorem 6
>  (dealing with number fields). Theorems
>  7 and 8 (concerning
>  $ax^n+bx^m+c\in\mathbb Z[x])$ are
>  proved by using the corresponding
>  theorems for rational function fields
>  together with a lemma which may be
>  viewed as a refinement of Hilbert's
>  irreducibility theorem. The proof of
>  this lemma is based on Siegel's
>  theorem (on integral points of curves
>  of positive genus) and on a result of
>  Maillet (1919) dealing with rational
>  functions over $\mathbb Q$ taking
>  infinitely many integral values at
>  rational points.
> 
> 
> {Reviewer's remarks: In Theorem 2 the
>  term $u^{\nu-\mu}$ in the expression
>  for $B$ has to be replaced by $u^\nu$.
>  The proof of Lemma 27 employs Lemma
>  2(c) although this lemma only applies
>  to separable extensions. In order to
>  prove Lemma 49 one has to know that
>  every finite separable extension $L$
>  of $K(t)$ with $L\subseteq \overline
> > K(t)$ is contained in $K'(t)$ for some
>  separable extension $K'$ of $K$. (One
>  can in fact prove that $L=K'(t)$ for
>  suitable $K'$. This need not be true
>  for inseparable $L$.) The proof of
>  Theorem 6 is apparently based on the
>  incorrect assumption that a divisor
>  $P$ of a function field $L=K(t,y)$ has
>  degree $1$ or is ramified with respect
>  to $K(t)$ if $t$ and $y$ are congruent
>  to elements of $K\bmod P$.} 
> 
> 
> REVISED (1995)
> 
> 
> Reviewed by G. Turnwald
> 
> 
> 


