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\author{Lizzy Hunt}
\date{\today}
\title{Assignment Five}
\hypersetup{
 pdfauthor={Lizzy Hunt},
 pdftitle={Assignment Five},
 pdfkeywords={},
 pdfsubject={},
 pdfcreator={Emacs 28.2 (Org mode 9.6.1)}, 
 pdflang={English}}
\begin{document}

\maketitle
\setlength\parindent{0pt}

\section{Section 4.1}
\label{sec:orge1c762a}
\subsection{Question One}
\label{sec:org6594d20}
\subsubsection{d}
\label{sec:orgf9d77b6}
\(2x^5 + x^4 + 6x^2 + 3x + 2\)

\subsection{Question Three}
\label{sec:org8e6afc5}
\subsubsection{b}
\label{sec:org7f9f78d}
\begin{verbatim}
>>> list(filter(lambda x: x, \
    [f"{a}x^2 + {b}x + {c}" if not a == 0 or not b == 0 else "" \
    for a in range(3) for b in range(3) for c in range(3)]))
\end{verbatim}

\{'0x\textsuperscript{2} + 1x + 0',
 '0x\textsuperscript{2} + 1x + 1',
 '0x\textsuperscript{2} + 1x + 2',
 '0x\textsuperscript{2} + 2x + 0',
 '0x\textsuperscript{2} + 2x + 1',
 '0x\textsuperscript{2} + 2x + 2',
 '1x\textsuperscript{2} + 0x + 0',
 '1x\textsuperscript{2} + 0x + 1',
 '1x\textsuperscript{2} + 0x + 2',
 '1x\textsuperscript{2} + 1x + 0',
 '1x\textsuperscript{2} + 1x + 1',
 '1x\textsuperscript{2} + 1x + 2',
 '1x\textsuperscript{2} + 2x + 0',
 '1x\textsuperscript{2} + 2x + 1',
 '1x\textsuperscript{2} + 2x + 2',
 '2x\textsuperscript{2} + 0x + 0',
 '2x\textsuperscript{2} + 0x + 1',
 '2x\textsuperscript{2} + 0x + 2',
 '2x\textsuperscript{2} + 1x + 0',
 '2x\textsuperscript{2} + 1x + 1',
 '2x\textsuperscript{2} + 1x + 2',
 '2x\textsuperscript{2} + 2x + 0',
 '2x\textsuperscript{2} + 2x + 1',
 '2x\textsuperscript{2} + 2x + 2'\}

\subsection{Question Five}
\label{sec:orgf501bb3}
Unfortunately, latex'ing long division is not trivial. Sorry in advance.

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\subsection{Question Six}
\label{sec:orgf9d6188}
\subsubsection{c}
\label{sec:org0feefbc}
No, since two polynomials of degree \(\leq k\) with \(k=2\), under multiplication, could produce
a polynomial of degree 4.
\subsubsection{d}
\label{sec:org18fd125}
\begin{itemize}
\item Closed under addition (two polynomials with even degrees can only add to polynomials with even degrees)
\item Closed under multiplication (two polynomials with even degrees can only multiply to polynomials with even degrees)
\item 0\textsubscript{R} exists in the set
\item For a polynomial \(a\) with only even powers, then \(a + x = 0_R\) implies that \(a = -x\) which will only change coefficients
\end{itemize}

Seems like a subring (although not rigorously shown) to me!

\subsubsection{e}
\label{sec:org6de4719}
Not a subring, since \(x^3 \cdot x = x^4\), with \(x^3\) and \(x\) both elements in the described set.
\subsection{Question Eleven}
\label{sec:orgb3179d8}

\begin{verbatim}
>>> list(filter(lambda x: ((3 + x) % 9 == 0) and ((3 * x) % 9 == 0), range(9)))
[6]
\end{verbatim}

\((1 + 3x)(1 + 6x) = 1 + 6x + 3x + 18x^2 = 1 + 9x + 18x^2 = 1\)
\end{document}