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i am interested in general criteria about even degree polynomial factorization,maybe number theories rule or some other mathematical properties can be helpful,i am talking rules how to factorize even degree polynomials or in other word if it is possible given polynomial with degree $(2,4,6,8,...2*n)$ could be factorized.of course zero degree polynomial

$p(x)=a_0$

can be always factorized with two product of constants,also pay attention that my question should satisfy following criteria :polynomial with degree $2*k$ should be factorized as product of two polynomial with degree $k$,let say

$a*x^2+b*x+c$

degree here is $2$,so each member in product should have degree $2/2=1$ or

we should solve following equation

$(k*x+m)(d*x+e)=a*x^2+b*x+c$

we may solve this if we enter some values of $a,b,c$,but what is relationship between coefficients of small degree polynomial and high one so that there should be quarantine of being solvable? thanks in advance

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Since you mentioned number theory I assume you are asking for factorization within the rational field. In general one has the Eisenstein criterion. It works surprisingly well. The algorithm implemented in maple, mathematica, etc may be also worth looking. I believe they do some sort of search algorithm over all smaller degree integer coefficient polynomials and perform long division into the polynomial. –  John Jiang Feb 24 at 5:44
    
but if we consider in general,introducing irrational or even complex terms?thanks for reply –  dato datuashvili Feb 24 at 5:47
    
Check the Fundamental Theorem of Algebra, the version of Carl Friedrich Gauß from 1799 hat your intended factorization as its title, every polynomial with real coefficients can be factored into linear and quadratic factors with real coefficients. –  LutzL Feb 24 at 6:55
    
ok thanks very much –  dato datuashvili Feb 24 at 8:27

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