\documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amssymb} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2570} %TCIDATA{} %TCIDATA{Created=Thursday, February 21, 2002 14:53:39} %TCIDATA{LastRevised=Monday, October 03, 2005 09:55:12} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{AllPages= %F=36,\PARA{038
Scientific Notebook: On Line Mathematics\dotfill \thepage} %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \section{Polyn\'{o}my} \subsection{D\^{o}kaz vety o racion\'{a}lnych kore\v{n}och polyn\'{o}mov s celo\v{c}\'{\i}seln\'{y}mi koeficintami} \newline Hodnota polyn\'{o}mu $f$ v \v{c}\'{\i}sle $\frac{p}{q}$ je nula, preto% \[ 0=a_{n}\left( \frac{p}{q}\right) ^{n}+a_{n-1}\left( \frac{p}{q}\right) ^{n-1}+\cdots +a_{1}\frac{p}{q}+a_{0} \] \[ 0=a_{n}\frac{p^{n}}{q^{n}}+a_{n-1}\frac{p^{n-1}}{q^{n-1}}+\cdots +a_{1}\frac{% p}{q}+a_{0}\ \left/ \cdot q^{n}\right. \]% \[ 0=a_{n}p^{n}+a_{n-1}p^{n-1}q+\cdots +a_{1}pq^{n-1}+a_{0}q^{n} \]% \[ -a_{n}p^{n}=a_{n-1}p^{n-1}q+\cdots +a_{1}pq^{n-1}+a_{0}q^{n} \]% \[ -a_{n}p^{n}=q\left( a_{n-1}p^{n-1}+\cdots +a_{1}pq^{n-2}+a_{0}q^{n-1}\right) \]% V\v{s}etky \v{c}\'{\i}sla v poslednej rovnosti s\'{u} cel\'{e}, preto z tejto rovnosti vypl\'{y}va, \v{z}e $q$ del\'{\i} $a_{n}p^{n}$ a ke\v{d}\v{z}% e $p$ a $q$ s\'{u} nes\'{u}delite\v{l}n\'{e}, mus\'{\i} $q$ deli\v{t} $a_{n}$% . Ak z tretej z predch\'{a}dzaj\'{u}cich rovn\'{\i}c vyjadr\'{\i}me $% a_{0}q^{n}$, dostaneme \[ -a_{0}q^{n}=p\left( a_{n}p^{n-1}+a_{n-1}p^{n-2}q+\cdots +a_{1}q^{n-1}\right) \]% z \v{c}oho vypl\'{y}va $p\mid a_{0}q^{n}$ a ke\v{d}\v{z}e $p$ a $q$ s\'{u} nes\'{u}delite\v{l}n\'{e}, tak $p\mid a_{0}$.\ $\blacksquare $ \begin{center} \begin{tabular}{|c|} \hline \textbf{% %TCIMACRO{\hyperref{Sp\"{a}\v{t}}{}{}{A53.tex#2}}% %BeginExpansion \msihyperref{Sp\"{a}\v{t}}{}{}{A53.tex#2}% %EndExpansion } \\ \hline \end{tabular} \end{center} \end{document}