A property of polynomials

$A P R O P E R T Y O F P O L Y N O M I A L S$

Let $f(x)$ be a polynomial in $x$ where $x$ $a n d t h e c o e f f i c i e n t s$ $a)areintegers,$ b) $\hskip 36.135pt\,$ x $i s a s q u a r e m a t r i x a n d$ k ${\hskip 0.0pt.5in}\ isanaturalnumber.$ $T h e p r o p e r t y :$ $f(x+k*f(x))iscongruentto$ 0 $m o d$ f(x) $.$ Proof:

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