# A property of polynomials

$APROPERTYOFPOLYNOMIALS$

Let $f(x)$             be a polynomial in $x$ where $x$$andthecoefficients$ $a)areintegers,$ b) $\hskip 36.135pt\,$ x $isasquarematrixand$k${\hskip 0.0pt.5in}\ isanaturalnumber.$ $Theproperty:$ $f(x+k*f(x))iscongruentto$0$mod$f(x)$.$Proof:

 $\hskip 36.135pt\\hskip 36.135pt\ Thereisnolossofgeneralityintakingk=1.% ByTaylor^{\prime}stheoremweget\par f(x+f(x))=f(x)+f(x)f^{\prime}(x)+(f(x)^{2}% )/2f^{\prime\prime}(x)+(f(x)^{3})/3!f^{\prime\prime\prime}(x)....=f(x){1+f^{% \prime}(x)+f(x)/2*f^{\prime\prime}(x)+f(x)^{2}/3!..}i.e.f(x+f(x))iscongruentto% 0(mod(f(x))).\par \begin{flushright}\begin{tabular}[]{|ll|}\hline Title&A % property of polynomials\\ Canonical name&APropertyOfPolynomials\\ Date of creation&2013-03-22 19:34:45\\ Last modified on&2013-03-22 19:34:45\\ Owner&akdevaraj (13230)\\ Last modified by&akdevaraj (13230)\\ Numerical id&17\\ Author&akdevaraj (13230)\\ Entry type&Definition\\ Classification&msc 11C08\\ \hline\end{tabular}\end{flushright}\end{document}$